AIML CA2

Time Series Analysis

Name: Glenn Wu Adm. No: 2214395 Class: DAAA/FT/2A/01

ARIMA and SARIMAX use different fitting methods https://stats.stackexchange.com/questions/374845/python-arima-generates-different-predictions-than-sarimax-for-same-orders

end_date = "2023-07-01" # 6 months from January 2023

import warnings
warnings.filterwarnings("ignore")

import pickle

import statsmodels.api as sm
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

import statsmodels

from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, mean_absolute_error, mean_absolute_percentage_error, r2_score
from sklearn.preprocessing import PowerTransformer
from scipy.stats import boxcox
from statsmodels.tsa.seasonal import seasonal_decompose
# from pmdarima.arima import CHTest, nsdiffs

from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

from statsmodels.tsa.stattools import q_stat, acf, pacf

# Custom date parsing function to get M/D/Y
def custom_date_parser(date_string):
    day, month, year = date_string.split('/')
    
    return pd.to_datetime(f'{day}/{month}/{year}', format='%d/%m/%Y')

df = pd.read_csv("Energy Consumption Dataset.csv", parse_dates=['DATE'], date_parser=custom_date_parser).set_index("DATE")
df.index = pd.DatetimeIndex(df.index, freq='MS')

Exploratory Data Analysis

We will begin by conducting an exploratory data analysis of the data, to gain a better understanding of the characteristics of the dataset.

The dataset is about Energy Consumption, likely collected by a utilities company. There are 397 rows and 2 columns.
Gas Consumption (tons): Gas consumption in tons
Electricity Consumption (MWh): Electricity consumption in MWh
Water Consumption (tons): Water consumption in tons

eda_df = df.copy()
eda_df.head(10)

	Gas Consumption (tons)	Electricity Consumption (MWh)	Water Consumption (tons)
DATE
1990-01-01	18.0	725.1	548.8
1990-02-01	15.8	706.7	640.7
1990-03-01	17.3	624.5	511.1
1990-04-01	18.9	574.7	515.3
1990-05-01	22.0	553.2	488.4
1990-06-01	23.4	580.9	503.2
1990-07-01	24.6	626.2	448.3
1990-08-01	24.8	632.5	411.7
1990-09-01	23.8	605.8	440.8
1990-10-01	20.2	563.2	579.0

eda_df.describe()

	Gas Consumption (tons)	Electricity Consumption (MWh)	Water Consumption (tons)
count	397.000000	397.000000	397.000000
mean	23.785139	888.472544	484.953652
std	4.903452	153.877594	133.908863
min	11.600000	553.200000	44.400000
25%	20.200000	771.100000	384.400000
50%	23.500000	897.800000	487.400000
75%	27.900000	1005.200000	580.200000
max	46.000000	1294.000000	811.000000

eda_df.dtypes

Gas Consumption (tons)           float64
Electricity Consumption (MWh)    float64
Water Consumption (tons)         float64
dtype: object

# Check missing values
eda_df.isnull().sum()

Gas Consumption (tons)           0
Electricity Consumption (MWh)    0
Water Consumption (tons)         0
dtype: int64

No missing values to fill

# Get the column names of numeric columns
numeric_columns = df.select_dtypes(include=[int, float]).columns.tolist()

numeric_columns

['Gas Consumption (tons)',
 'Electricity Consumption (MWh)',
 'Water Consumption (tons)']

sns.set()

fig, axes = plt.subplots(nrows=len(numeric_columns), ncols=1, figsize=(16, 15))

for i, col in enumerate(numeric_columns):
    sns.lineplot(data=eda_df.reset_index(), x="DATE", y=col, ax=axes[i])
plt.show()

Gas Consumption

gas_consumption_series = df["Gas Consumption (tons)"]
electricity_consumption_series = df["Electricity Consumption (MWh)"]
water_consumption_series = df["Water Consumption (tons)"]

from scipy import stats

def visualisations(series, label="Time Series Data"):

    fig, axes = plt.subplots(ncols=2, nrows=1, figsize=(16, 5))

    # Histogram
    sns.histplot(series, bins=20, kde=True, ax=axes[0])
    axes[0].set_title(f'Histogram of {label}')
    axes[0].set_xlabel('Value')
    axes[0].set_ylabel('Frequency')
    axes[0].set_xlabel(f"{label} (Skewness: {series.skew()})")

    # Q-Q plot
    stats.probplot(series, dist='norm', plot=axes[1])
    axes[1].set_title('Q-Q Plot')
    axes[1].set_xlabel('Theoretical Quantiles')
    axes[1].set_ylabel('Sample Quantiles')
    plt.show()

visualisations(gas_consumption_series, "Gas Consumption")
visualisations(electricity_consumption_series, "Electricity Consumption")
visualisations(water_consumption_series, "Water Consumption")

Observations

• All the time series have a similar distribution, based on QQ Plot
• The times series all have distributions that are neearly normal distributions
• Low (<0.5) skew

class NullTransformer:
    def __init__(self):
        return
    def fit_transform(self, X):
        return X
    def inverse_transform(self, X):
        return X
    
class LogTransformer:
    def __init__(self):
        return
    def fit_transform(self, X):
        return np.log(X)
    def inverse_transform(self, X):
        return np.exp(X)

pt = NullTransformer()

def transform_series(series, transformer):
    a = transformer.fit_transform(np.array(series).reshape(-1, 1))
    x = pd.Series([x[0] for x in a])
    x.index = series.index
    return x

def reverse_transform_series(series, transformer):
    a = transformer.inverse_transform(np.array(series).reshape(-1, 1))
    x = pd.Series([x[0] for x in a])
    x.index = series.index
    return x


b = transform_series(gas_consumption_series, pt)
reverse_transform_series(b, pt)

DATE
1990-01-01    18.0
1990-02-01    15.8
1990-03-01    17.3
1990-04-01    18.9
1990-05-01    22.0
              ... 
2022-09-01    27.7
2022-10-01    31.8
2022-11-01    31.0
2022-12-01    32.4
2023-01-01    31.3
Freq: MS, Length: 397, dtype: float64

Yeo-Johnson Transformation

$ Y(\lambda) = \begin{cases} \left((1 + \lambda) \cdot Y + 1\right)^{\frac{1}{\lambda}} - 1, & \text{if } \lambda \neq 0 \text{ and } Y \geq 0, \\ \log(1 + Y), & \text{if } \lambda = 0 \text{ and } Y \geq 0, \\ -\left((-Y + 1)^{2 - \lambda} - 1\right), & \text{if } \lambda \neq 2 \text{ and } Y < 0, \\ -\log(-Y + 1), & \text{if } \lambda = 2 \text{ and } Y < 0, \end{cases} $

where:

• $Y$ is the original data value.
• $\lambda$ is the transformation parameter that determines the type and amount of transformation. It can take any real value, including zero.

Time series prediction methods like SARIMA and VARMA assume that the residuals are normally distributed. Transforming the data using the Yeo-Johnson transformation can help achieve a more normal distribution.

The transformed values are usually more symmetric and have more stable variance compared to the original data.

gas_pt = PowerTransformer()
electricity_pt = PowerTransformer()
water_pt = PowerTransformer()

# gas_pt = NullTransformer()
# electricity_pt = NullTransformer()
# water_pt = NullTransformer()

# gas_pt = LogTransformer()
# electricity_pt = LogTransformer()
# water_pt = LogTransformer()

gas_consumption_series = transform_series(gas_consumption_series, gas_pt)
electricity_consumption_series = transform_series(electricity_consumption_series, electricity_pt)
water_consumption_series = transform_series(water_consumption_series, water_pt)

visualisations(gas_consumption_series, "Yeo-Johnson Transformed Gas Consumption")
visualisations(electricity_consumption_series, "Yeo-Johnson Transformed Electricity Consumption")
visualisations(water_consumption_series, "Yeo-Johnson Transformed Water Consumption")

Pearson's r Correlations

To check for correlation between three series, we use Pearson's r correlation coefficient. It is able to measure the linear correlation between the two variables.

The r value of X and Y is equal to diving the covriance of X and Y by the product of their standard deviations.

Formula: \begin{equation} \rho = \frac{\text{cov}(X,Y)}{\sigma_x \sigma_y} \end{equation}

\begin{equation} r = \frac{{}\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})} {\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2(y_i - \overline{y})^2}} \end{equation}

# Correlation matrix
plt.figure(figsize=(12, 8))
sns.heatmap(eda_df.corr(), annot=True, cmap='coolwarm')
plt.title('Correlation Heatmap')
plt.show()

def seasonal_diff(series, seasonal_period=12, return_discard=False):
    x = np.array(series[seasonal_period:])-np.array(series[:-seasonal_period])
    x = pd.Series(x)
    x.index = series[seasonal_period:].index
    if return_discard:
        return x, series[:seasonal_period]
    else:
        return x
    
def seasonal_diff_order(series, order, seasonal_period=12, return_discards=True):
    x = series
    discards = []
    for i in range(order):
        x, discard = seasonal_diff(x, seasonal_period=seasonal_period, return_discard=True)
        discards.append(discard)
    if return_discards:
        return x, discards
    else:
        return x
    
def reverse_seasonal_diff(diffed, discard, seasonal_period=12):
    x = discard.append(diffed)
    for i in range(seasonal_period):
        x[i::seasonal_period] = np.cumsum(x[i::seasonal_period])
    return x

def reverse_seasonal_diff_order(diffed, discards, seasonal_period=12):
    x = reverse_seasonal_diff(diffed, discards[-1], seasonal_period=seasonal_period)
    discards = discards[:-1]
    if discards == []:
        return x
    else:
        return reverse_seasonal_diff_order(x, discards, seasonal_period=seasonal_period)
    
# find the combination of differencing orders that minimizes standard deviation
def find_best_diff_order(series, max_d=3, max_D=3, max_M=24, seasonal_period=12):
    # maximum d+D = 2
    scores = []
    for d in range(max_d):
        for D in range(max_D):
            for M in range(2, max_M):
                scores.append(np.std(seasonal_diff_order(seasonal_diff_order(series, d, seasonal_period=1, return_discards=False), D, seasonal_period=M, return_discards=False)))
#                 print(scores[-1], d, D, M)
    dD, M = np.divmod(np.argmin(scores), max_M-2)
    M += 2
    return np.divmod(dD, max_D), M

def k_diff(series, d, D, seasonal_period=12, return_discards=False):
    normal_diff_series, normal_discards = seasonal_diff_order(series, d, seasonal_period=1, return_discards=True)
    seasonal_diff_series, seasonal_discards = seasonal_diff_order(normal_diff_series, D, seasonal_period=seasonal_period, return_discards=True)
    if return_discards:
        return seasonal_diff_series, normal_discards, seasonal_discards
    else: 
        return seasonal_diff_series
    
def k_undiff(series, normal_discards, seasonal_discards, seasonal_period=12):
    undiff_normal = reverse_seasonal_diff_order(series, normal_discards, seasonal_period=1)
    if seasonal_discards == None:
        return undiff_normal
    else:
        return reverse_seasonal_diff_order(undiff_normal, seasonal_discards, seasonal_period=seasonal_period)

differenced_eda_df = eda_df.copy()

differenced_eda_df["Gas Consumption (tons)"] = k_diff(differenced_eda_df["Gas Consumption (tons)"], 1, 0)
differenced_eda_df["Electricity Consumption (MWh)"] = k_diff(differenced_eda_df["Electricity Consumption (MWh)"], 1, 1)
differenced_eda_df["Water Consumption (tons)"] = k_diff(differenced_eda_df["Water Consumption (tons)"], 1, 0)

differenced_eda_df = differenced_eda_df.dropna()

# Correlation matrix
plt.figure(figsize=(12, 8))
sns.heatmap(differenced_eda_df.corr(), annot=True, cmap='coolwarm')
plt.title('Correlation Heatmap after Appropriate Differencing')
plt.show()

Observations

• Since Water and Gas consumption are moderately negatively correlated I will explore using a VARMAX model on them.
• Electricity consumption has a very low, near zero, correlation with these columns after differencing for stationarity.

Statistical Test for Causality

---

Granger Causality Test

Hypotheses:
$H_0$: The lagged values of variable X do not cause the values of variable Y.

$H_A$: The lagged values of variable X do Granger cause the values of variable Y.

from statsmodels.tsa.stattools import grangercausalitytests

maxlag=30
test = 'ssr_chi2test'

def grangers_causation_matrix(data, variables, test='ssr_chi2test', verbose=False):
    df = pd.DataFrame(np.zeros((len(variables), len(variables))), columns=variables, index=variables)
    for c in df.columns:
        for r in df.index:
            test_result = grangercausalitytests(data[[r, c]], maxlag=maxlag, verbose=False)
            p_values = [round(test_result[i+1][0][test][1], 4) for i in range(maxlag)]
            if verbose:
                print(f'Y = {r}, X = {c}, P Values = {p_values}')
            df.loc[r, c] = np.min(p_values)
    df.columns = [var + '_x' for var in variables]
    df.index = [var + '_y' for var in variables]
    return df

plt.figure(figsize=(12, 8))
sns.heatmap(grangers_causation_matrix(differenced_eda_df, variables = differenced_eda_df.columns), annot=True, cmap='coolwarm')
plt.title('Granger Causation Heatmap')
plt.show()

Based on a 0.05 significance level, we can conclude that Gas Consumption causes Water Consumption and Electricity Consumption causes Gas Consumption.

Seasonality

Seasonality in the time series can be used to enhance predictions. There are many methods to determine a suitable seasonal period. The first method I will use is to use the seasonal_decompose() function to find the seasonal component for a range of seasonal periods then calculate the standard deviation of this component. The standard deviation of a component at a suitable period should tend to be greater as it makes up a greater part of the time series.

def find_seasonal_period(series, title="", period_range=range(2, 50), **kwargs):
    plt.figure(**kwargs)
    # ranges = []
    seasonal_stds = []
    residual_stds = []
    for i in period_range:
        decompose_data = seasonal_decompose(series, model="add", period=i)
    #     ranges.append(np.max(decompose_data.seasonal)-np.min(decompose_data.seasonal))
        seasonal_stds.append(np.std(decompose_data.seasonal[:i]))
        residual_stds.append(np.std(decompose_data.resid.dropna()))
    sns.lineplot(x=list(period_range), y=seasonal_stds)
    if title != "":
        plt.title(title)
    plt.xlabel(f"Seasonal period with the greatest standard deviation of seasonal component: {list(period_range)[np.argmax(seasonal_stds)]}")
    plt.show()
#     print()

find_seasonal_period(gas_consumption_series, title="Standard Deviations of the Seasonal Component for each Seasonal Period, Gas Consumption", figsize=(10, 5))
find_seasonal_period(electricity_consumption_series, title="Standard Deviations of the Seasonal Component at each Seasonal Period, Electricity Consumption", figsize=(10, 5))
find_seasonal_period(water_consumption_series, title="Standard Deviations of the Seasonal Component at each Seasonal Period, Water Consumption", figsize=(10, 5))

Observations

• For Gas and Water consumption, it is difficult to see any trend and it generally seems like standard deviation increases with the seasonal period.
• For Electricity consumption, there is an obvious trend where seasonal periods that are multiples of 6 (eg. 12, 18, 24) have a much greater standard deviation. Multiples of 12 seem to have an even greater standard deviation, thus I will use 12 as the seasonal period for this series.

def seasonal_decompose_plot(series, title, **kwargs):
    decompose_data = seasonal_decompose(series, **kwargs)
    x = decompose_data.plot()
    x.set_size_inches(16, 10)
    x.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9, wspace=0.2, hspace=0.2)
    x.suptitle(title, fontsize=18)
    plt.show()
    return decompose_data

gas_consumption_seasonal_decompose = seasonal_decompose_plot(gas_consumption_series, "Gas Consumption", period=48)
electricity_consumption_seasonal_decompose = seasonal_decompose_plot(electricity_consumption_series, "Electricity Consumption", period=12)
water_consumption_seasonal_decompose = seasonal_decompose_plot(water_consumption_series, "Water Consumption", period=45)

Statistical Tests for Stationarity

---

Augmented Dickey-Fuller Test

Hypotheses:
$H_0$: The data is non-stationary.

$H_A$: The data is stationary.

Kwiatkowski Phillips Schmidt Shin Test

Hypotheses:
$H_0$: The data is stationary.

$H_A$: The data is not stationary.

from statsmodels.tsa.stattools import adfuller
from statsmodels.tsa.stattools import kpss

def stationarity_tests(series, a=0.05):
    warnings.filterwarnings("ignore")
    

    # https://www.statology.org/dickey-fuller-test-python/
    # https://analyticsindiamag.com/how-to-make-a-time-series-stationary/
    # test whether the data is stationary or non-stationary using Augmented Dickey-Fuller Test and Kwiatkowski Phillips Schmidt Shin Test

#     print(f"Augmented Dickey-Fuller Test\nHypotheses:\n{'-'*20}\nH0: The data is non-stationary. \nHA: The data is stationary.\n{'-'*20}")

    adf_test = adfuller(series)

    (adf_test_statistic, adf_p_value) = adf_test[0: 2]
#     print(f"Test Statistics: {test_statistic}")
#     print(f"P-Value: {p_value}")
#     if p_value < 0.05:
#         print(f"{'-'*20}\nSince P-value < significance level of 0.05, we can reject the null hypothesis and conclude that the data is stationary.")
#     else:
#         print(f"{'-'*20}\nSince P-value > significance level of 0.05, we cannot reject the null hypothesis and thus conclude that the data is not stationary.")
    
#     print(f"\nKwiatkowski Phillips Schmidt Shin Test\nHypotheses:\n{'-'*20}\nH0: The data is stationary.\nHA: The data is not stationary.\n{'-'*20}")

    kpss_test = kpss(series, regression="ct")

    (kpss_test_statistic, kpss_p_value) = kpss_test[0: 2]
#     print(f"Test Statistics: {test_statistic}")
#     print(f"P-Value: {p_value}")
#     if p_value < 0.05:
#         print(f"{'-'*20}\nSince P-value < significance level of 0.05, we can reject the null hypothesis and conclude that the data is not stationary.")
#     else:
#         print(f"{'-'*20}\nSince P-value > significance level of 0.05, we cannot reject the null hypothesis and thus conclude that the data is stationary.")
 
    results_df = pd.DataFrame(columns=["Test Statistic", "P-Value", "Conclusion"])
    
    results_df["Test Statistic"] = [adf_test_statistic, kpss_test_statistic]
    results_df["P-Value"] = [adf_p_value, kpss_p_value]
    results_df["Conclusion"] = ["Stationary" if adf_p_value < a else "Non-Stationary", "Non-Stationary" if kpss_p_value < a else "Stationary"]
    
    results_df.index = ["Augmented Dickey-Fuller Test", "Kwiatkowski-Phillips-Schmidt-Shin-Test"]
    
    print(f"At {a} significance level:")
    
    return results_df
    
    
    # InterpolationWarning: The test statistic is outside of the range of p-values available in the
    # look-up table. The actual p-value is smaller than the p-value returned. <-- ***

Cases

1. Both tests conclude that the given series is stationary – The series is stationary
2. Both tests conclude that the given series is non-stationary – The series is non-stationary
3. ADF concludes non-stationary, and KPSS concludes stationary – The series is trend stationary. To make the series strictly stationary, the trend needs to be removed in this case. Then the detrended series is checked for stationarity.
4. ADF concludes stationary, and KPSS concludes non-stationary – The series is difference stationary. Differencing is to be used to make series stationary. Then the differenced series is checked for stationarity.
   https://www.analyticsvidhya.com/blog/2021/06/statistical-tests-to-check-stationarity-in-time-series-part-1/

print("Gas Consumption Series:")

stationarity_tests(gas_consumption_series)

Gas Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-3.418406	0.010347	Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.296366	0.010000	Non-Stationary

print("Electricity Consumption Series:")

stationarity_tests(electricity_consumption_series)

Electricity Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-2.195499	0.207883	Non-Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.485181	0.010000	Non-Stationary

print("Water Consumption Series:")

stationarity_tests(water_consumption_series)

Water Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-4.676428	0.000093	Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.274685	0.010000	Non-Stationary

Observation

• Gas Consumption Series: falls under Case 4, differencing is needed to make the data strictly stationary
• Electricity Consumption Series: falls under Case 2
• Water Consumption Series: falls under Case 4, differencing is needed to make the data strictly stationary

Differencing

The ARIMA family of time series models have a built in differencing $d$ component, where $d$ is the differencing order.
https://people.duke.edu/~rnau/411arim2.htm

From the article:

• Rule 1: If the series has positive autocorrelations out to a high number of lags, then it probably needs a higher order of differencing.
• Rule 2: If the lag-1 autocorrelation is zero or negative, or the autocorrelations are all small and patternless, then the series does not need a higher order of differencing. If the lag-1 autocorrelation is -0.5 or more negative, the series may be overdifferenced.
• Rule 3: The optimal order of differencing is often the order of differencing at which the standard deviation is lowest.

One of the methods to choose an appropriate differencing term is by looking at the Autocorrelation and Partial Autocorrelation Plots.

def differencing_acf_pacf_plots(series, plot_log=True, lags=50):

    fig, axes = plt.subplots(ncols=2, nrows=1, figsize=(16, 5))

    plot_acf(series, ax=axes[0], lags=lags, alpha=0.5, label="Default")
    plot_pacf(series, ax=axes[1], lags=lags, method='ywm', alpha=0.5, label="Default")
    
#     if plot_log:
#         plot_acf(np.log(series), ax=axes[0], lags=20, alpha=0.5, label="Log Transform")
#         plot_pacf(np.log(series), ax=axes[1], lags=20, method='ywm', alpha=0.5, label="Log Transform")
#         axes[0].legend(["Default", "Log Transform"])
#         axes[0].get_legend().legendHandles[0].set_color((0.2980392156862745, 0.4470588235294118, 0.6901960784313725))
#         axes[0].get_legend().legendHandles[1].set_color((0.8666666666666667, 0.5176470588235295, 0.3215686274509804))

#         axes[1].legend(["Default", "Log Transform"])
#         axes[1].get_legend().legendHandles[0].set_color((0.2980392156862745, 0.4470588235294118, 0.6901960784313725))
#         axes[1].get_legend().legendHandles[1].set_color((0.8666666666666667, 0.5176470588235295, 0.3215686274509804))

    plt.show()

print("Gas Consumption Series:\n")
    
print("No differencing:")
differencing_acf_pacf_plots(gas_consumption_series)

# x = np.log(gas_consumption_series)
x = gas_consumption_series
print("First order differencing:")
differencing_acf_pacf_plots(np.diff(x), plot_log=False)

Gas Consumption Series:

No differencing:

First order differencing:

Observation
$d=1$ is the appropriate differencing term.

Differencing

Another method to find the appropriate differencing term as well as the appropriate seasonal differencing term, $D$, is by evaluating the standard deviation of the resulting differenced series. The optimal differencing terms will result in the lowest standard deviation. This method also finds the optimal seasonal period if there is a seasonal differencing term.

print(f"Gas Consumption Series Differencing Orders: {find_best_diff_order(gas_consumption_series)}")
print(f"Electricity Consumption Series Differencing Orders: {find_best_diff_order(electricity_consumption_series)}")
print(f"Water Consumption Series Differencing Orders: {find_best_diff_order(water_consumption_series)}")

Gas Consumption Series Differencing Orders: ((1, 0), 2)
Electricity Consumption Series Differencing Orders: ((1, 1), 12)
Water Consumption Series Differencing Orders: ((1, 0), 2)

fig, axes = plt.subplots(nrows=len(numeric_columns), ncols=1, figsize=(16, 15))

for i, col in enumerate(numeric_columns):
    sns.lineplot(data=differenced_eda_df.reset_index(), x="DATE", y=col, ax=axes[i])
plt.show()

print("Gas Consumption Series:")

stationarity_tests(k_diff(gas_consumption_series, 1, 0))

Gas Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-7.811089	7.066742e-12	Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.089844	1.000000e-01	Stationary

print("Electricity Consumption Series:")

stationarity_tests(k_diff(electricity_consumption_series, 1, 1, seasonal_period=12))

Electricity Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-8.019366	2.096457e-12	Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.048818	1.000000e-01	Stationary

print("Water Consumption Series:")

stationarity_tests(k_diff(water_consumption_series, 1, 0))

Water Consumption Series:
At 0.05 significance level:

	Test Statistic	P-Value	Conclusion
Augmented Dickey-Fuller Test	-9.062086	4.534434e-15	Stationary
Kwiatkowski-Phillips-Schmidt-Shin-Test	0.095551	1.000000e-01	Stationary

Model Evaluation

One of the goals set out by the assignment is 2. Tune the hyperparameters of the time series model to maximize the accuracy for training data and testing data prediction. Thus, I will select my evaluation metrics accordingly.

Metrics:

1. AIC (Akaike Information Criterion)

   • Formula: $\text{AIC} = 2k - 2\ln(\hat{L})$

   • Legend:

     • $k$: Number of estimated parameters in the model.

     • $\hat{L}$: Maximum likelihood estimation of the model.

       
       

2. RMSE (Root Mean Square Error)

   • Formula: $\text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}$

3. MAE (Mean Absolute Error)

   • Formula: $\text{MAE} = \frac{1}{n}\sum_{i=1}^{n}|y_i - \hat{y}_i|$

• AIC scores models based on fit and complexity (number of parameters). It is useful for model selection and preventing overfitting but does not account for forecast accuracy.
• RMSE and MAE measure the accuracy of forecasts, indicating how well the model performs in terms of predicting actual values, which is the goal of this task. I evaluate both of these metrics since the difference between the value of RMSE and MAE can indicate whether there are more or less outliers in the predictions.

from tqdm.notebook import tqdm

# def walkforward_validation(model, params, data, validation_size=0.2):
#     """  v1
#     model: function that returns a class with the fit() method which returns class with forecast() method
#     """
#     constant_validation_data = data[:round(len(data)*(1-validation_size))]
#     walking_validation_data = data[round(len(data)*(1-validation_size)):]
# #     print(constant_validation_data)
#     validation_sets = []
#     for i in range(len(walking_validation_data)-1):
#         datapoint = walking_validation_data[i:i+1]
# #         print(np.append(constant_validation_data, datapoint))
#         if i == 0:
#             validation_sets.append(constant_validation_data.append(datapoint))
#         else: 
#             validation_sets.append(validation_sets[i-1].append(datapoint))
    
#     predictions = pd.Series()
#     predictions.index = pd.DatetimeIndex(predictions.index, freq='MS')
    
#     for i in tqdm(range(len(validation_sets))):
# #         print(validation_sets[i])
#         sarimax_model = model(validation_sets[i], params)
#         forecast = sarimax_model.fit().forecast(steps=1)
#         predictions = predictions.append(forecast)
# #         print(forecast)
        
#     return walking_validation_data[1:], predictions

def walkforward_validation(model, params, data, rolling=True, gap=2, step=2, forecast_period=6, validation_size=0.1, flatten=True, return_train=False, return_coefficients=False, verbose=True
                          ):
    """ 
    walk-forward validation v2
    model: function that returns a class with the fit() method which returns class with forecast() method
    params: hyperparameters for model()
    data: time series to fit the model on
    rolling: whether to use rolling or expanding window
    forecast_period: number of months to predict using each set of training data by
    step: number of months to increase each consecutive set of training data by
    validation_size: fraction of dataset to use for validating predictions
    flatten: return the forecasts and predictions as 1d array or 2d array of size n_train_sets, forecast_period
    return_coefficients: whether to return the ar, ma, sar, sma coefficients of the model (disable for non-arima models)
    """

    forecast_period += gap
# model = statespace
# forecast_period = 20
# 6-month forecasting window for eg.
# step = 1
# data = gas_consumption_series
# validation_size=0.2
    constant_train_data = data[:round(len(data)*(1-validation_size))]
    train_data = data[round(len(data)*(1-validation_size)):]
    train_sets = [constant_train_data]
#     print(constant_train_data)
    for i in range(step, len(train_data)-forecast_period-step, step):
        train_sets.append(constant_train_data.append(train_data[:i])[i if rolling else 0:])
#         print(train_sets[-1)
#     print(len(train_sets))
    validation_sets = []
    for i in range(0, len(train_data)-forecast_period, step):
        validation_sets.append(train_data[i:i+forecast_period])

    forecast_sets = []
    
    if return_coefficients:
        model_parameters = []
    if return_train:
        in_sample_sets = []
    
    for i in tqdm(range(len(train_sets))) if verbose else range(len(train_sets)):
    #         print(validation_sets[i])
        tsa_model = model(train_sets[i], params)
#         print(train_sets[i].index)
#         print(len(train_sets[i]), len(train_sets[i]))
        tsa_model_fit = tsa_model.fit()
        forecast = tsa_model_fit.predict(start=len(train_sets[i]), end=len(train_sets[i])+forecast_period-1)
        forecast_sets.append(forecast)
        
        if return_train:
            in_sample = tsa_model_fit.predict()
            in_sample_sets.append(in_sample)
        
        if return_coefficients:
            model_parameters.append({})
            model_parameters[i]["ar"] = tsa_model_fit.polynomial_ar
            model_parameters[i]["ma"] = tsa_model_fit.polynomial_ma
            model_parameters[i]["sar"] = tsa_model_fit.polynomial_seasonal_ar
            model_parameters[i]["sma"] = tsa_model_fit.polynomial_seasonal_ma
#         print(forecast)

    returns = []
    
    if flatten:
        validations = validation_sets[0]
        forecasts = forecast_sets[0]
        for i in range(1, len(forecast_sets)):
            validations = validations.append(validation_sets[i])
            forecasts = forecasts.append(forecast_sets[i])
        returns.append(validations)
        returns.append(forecasts)
            
        if return_train:
            trains = train_sets[0]
            in_samples = in_sample_sets[0]
            for i in range(1, len(in_sample_sets)):
                trains = trains.append(train_sets[i])
                in_samples = in_samples.append(in_sample_sets[i])
            returns.append(trains)
            returns.append(in_samples)
    else:
        returns.append(validation_sets)
        returns.append(forecast_sets)
        
        if return_train:
            returns.append(train_sets)
            returns.append(in_sample_sets)
        
    if return_coefficients:
        returns.append(model_parameters)
    
#     print(train_sets[0])
#     print(in_sample_sets[0])
    return returns

# def holt(endog, params):
#     return statsmodels.tsa.holtwinters.Holt(endog, **params)

def hwes(endog, params):
    return statsmodels.tsa.holtwinters.ExponentialSmoothing(endog, **params)

def arima(endog, params):
    return sm.tsa.arima.ARIMA(endog, **params)

def sarimax(endog, params):
    return sm.tsa.statespace.SARIMAX(endog, **params)

def statespace(endog, params):
    return statsmodels.tsa.statespace.structural.UnobservedComponents(endog, **params)

def varmax(endog, params):
    return statsmodels.tsa.statespace.varmax.VARMAX(endog, **params)

def aic_score(model, params, series):
#     print(params)
    tsa_model = model(series, params)
    return -tsa_model.fit().aic

import copy

def arima_(endog, params):
    params = copy.deepcopy(params)
    params["order"] = (params.pop("p"), params.pop("d"), params.pop("q"))
    params["seasonal_order"] = (params.pop("P"), params.pop("D"), params.pop("Q"), params.pop("M"))
    return arima(endog, params)

def sarimax_(endog, params):
    params = copy.deepcopy(params)
    params["order"] = (params.pop("p"), params.pop("d"), params.pop("q"))
    params["seasonal_order"] = (params.pop("P"), params.pop("D"), params.pop("Q"), params.pop("M"))
    return sarimax(endog, params)

def varmax_(endog, params):
    params = copy.deepcopy(params)
    params["order"] = (params.pop("p"), params.pop("q"))
    if params["order"] == (0, 0):
        params["order"] = (1, 0)
    return varmax(endog, params)

# class dummy:
#     def __init__(self, endog, params):
#         self.endog = endog
#         self.params = params
#     def fit(self):
#         return dummyPredictWrapper(sum(self.endog)/len(self.endog))
        
# class dummyPredictWrapper:
#     def __init__(self, average):
#         self.result = average
#     def predict(start=0, end=0):
#         return [self.result] * (end-start)
    
# wf_valid, wf_pred, wf_train, wf_in_sample, coefs = walkforward_validation(dummy, {}, gas_consumption_series, validation_size=0.1, return_train=True)

Determining AR and MA terms

def plot_acf_pacf(series, d, D, seasonal_period=12, plot_seasonal=False):
    fig, axes = plt.subplots(ncols=1, nrows=2, figsize=(8, 10))
    
    lags = np.arange(seasonal_period, seasonal_period*8, seasonal_period) if plot_seasonal else 50

    plot_acf(k_diff(series, d, D, seasonal_period=seasonal_period), ax=axes[0], lags=lags, alpha=0.5, label="Default")
    plot_pacf(k_diff(series, d, D, seasonal_period=seasonal_period), ax=axes[1], lags=lags, method='ywm', alpha=0.5, label="Default")

    plt.show()

Gas Consumption Series

plot_acf_pacf(gas_consumption_series, 1, 0)

Observations:

• ACF somewhat cuts off at lag-1 so I will add an $MA$ term
• There isn't an obvious seasonal period from the original data so I will use only an ARIMA model instead of SARIMA first.

# params = {"order": (0, 1, 2), "seasonal_order": (1, 1, 1, 12), "enforce_stationarity": False, "enforce_invertibility": True}
# params = {"order": (0, 1, 2)}

def evaluation_descriptive(series, model, params, transformer, return_scores=False, plot_arima=True, is_hwes=False, return_coefficients=True, **kwargs):

    temp_arima = model(series, params).fit()
    print("AIC: ", temp_arima.aic)

    if return_coefficients:
        wf_valid, wf_pred, wf_train, wf_in_sample, coefs = walkforward_validation(model, params, series, validation_size=0.1, return_train=True, return_coefficients=True)
    else:
        wf_valid, wf_pred, wf_train, wf_in_sample = walkforward_validation(model, params, series, validation_size=0.1, return_train=True, return_coefficients=False)

    # print("R2 (walk forward validation): ", r2_score(*wf_valid))
#     print(wf_in_sample)
    scores = pd.DataFrame()
    
    scores["mae_test"] = [mean_absolute_error(reverse_transform_series(wf_valid, transformer), reverse_transform_series(wf_pred, transformer))]
    scores["mae_train"] = [mean_absolute_error(reverse_transform_series(wf_train, transformer), reverse_transform_series(wf_in_sample, transformer))] if not is_hwes else 0 
    scores["rmse_test"] = [mean_squared_error(reverse_transform_series(wf_valid, transformer), reverse_transform_series(wf_pred, transformer), squared=False)]
    scores["rmse_train"] = [mean_squared_error(reverse_transform_series(wf_train, transformer), reverse_transform_series(wf_in_sample, transformer), squared=False)] if not is_hwes else 0
    
    print("MAE (test): ", scores["mae_test"][0])
    print("MAE (train): ", scores["mae_train"][0])
    print("RMSE (test): ", scores["rmse_test"][0])
    print("RMSE (train): ", scores["rmse_train"][0])

    # gc_arima
    
#     return wf_valid, wf_pred

    if plot_arima:
        print("AR Coefficients: ", temp_arima.polynomial_ar)
        print("MA Coefficients: ", temp_arima.polynomial_ma)
        print("SAR Coefficients: ", temp_arima.polynomial_seasonal_ar)
        print("SMA Coefficients: ", temp_arima.polynomial_seasonal_ma)
        
        plot_acf_pacf(temp_arima.resid, 0, 0, **kwargs)
    
    if return_scores:
        return scores

evaluation_descriptive(gas_consumption_series, arima, {"order": (0, 1, 1)}, gas_pt)

AIC:  886.1087436643672

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  2.4073503629944932
MAE (train):  2.485491324638523
RMSE (test):  3.118468218704852
RMSE (train):  3.7108166291410996
AR Coefficients:  [1.]
MA Coefficients:  [ 1.         -0.53580746]
SAR Coefficients:  [1.]
SMA Coefficients:  [1.]

Based on rule 13 from this article, at lag-12 for example the autocorrelation is negative thus I will add an $SMA(1)$ term.
Since I am unsure of the seasonal period, I will implement a function to try different seasonal periods with their corresponding SAR or SMA term based on the sign at the lag.

n_seasonal_periods_range =  range(4, 40)
scores = []

# validation_gas_consumption_series = gas_consumption_series.iloc[:round(len(gas_consumption_series)*0.8)]

for seasonal_period in tqdm(n_seasonal_periods_range):
    autocorr = acf(k_diff(gas_consumption_series, 1, 0), nlags=seasonal_period)[-1]
    if autocorr > 0:
        P = 1
        Q = 0
    elif autocorr < 0:
        P = 0
        Q = 1
    else:
        P = 0
        Q = 0
    params = {"order": (0, 1, 1), "seasonal_order": (P, 0, Q, seasonal_period)}
    scores.append(mean_squared_error(*walkforward_validation(arima, params, gas_consumption_series), squared=False))
    print(params, scores[-1])
    
print(n_seasonal_periods_range[np.argmin(scores)])

  0%|          | 0/36 [00:00<?, ?it/s]

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (1, 0, 0, 4)} 0.6236267586521473

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (0, 0, 1, 5)} 0.6320317371174942

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (1, 0, 0, 6)} 0.6200251619663459

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (1, 0, 0, 7)} 0.6189338801352061

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (0, 0, 1, 8)} 0.6154111565252569

  0%|          | 0/15 [00:00<?, ?it/s]

{'order': (0, 1, 1), 'seasonal_order': (1, 0, 0, 9)} 0.6222808581711753

  0%|          | 0/15 [00:00<?, ?it/s]

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Input In [41], in <cell line: 6>()
     16         Q = 0
     17     params = {"order": (0, 1, 1), "seasonal_order": (P, 0, Q, seasonal_period)}
---> 18     scores.append(mean_squared_error(*walkforward_validation(arima, params, gas_consumption_series), squared=False))
     19     print(params, scores[-1])
     21 print(n_seasonal_periods_range[np.argmin(scores)])

Input In [36], in walkforward_validation(model, params, data, rolling, gap, step, forecast_period, validation_size, flatten, return_train, return_coefficients, verbose)
     74         tsa_model = model(train_sets[i], params)
     75 #         print(train_sets[i].index)
     76 #         print(len(train_sets[i]), len(train_sets[i]))
---> 77         tsa_model_fit = tsa_model.fit()
     78         forecast = tsa_model_fit.predict(start=len(train_sets[i]), end=len(train_sets[i])+forecast_period-1)
     79         forecast_sets.append(forecast)

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\arima\model.py:390, in ARIMA.fit(self, start_params, transformed, includes_fixed, method, method_kwargs, gls, gls_kwargs, cov_type, cov_kwds, return_params, low_memory)
    387 else:
    388     method_kwargs.setdefault('disp', 0)
--> 390     res = super().fit(
    391         return_params=return_params, low_memory=low_memory,
    392         cov_type=cov_type, cov_kwds=cov_kwds, **method_kwargs)
    393     if not return_params:
    394         res.fit_details = res.mlefit

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\statespace\mlemodel.py:728, in MLEModel.fit(self, start_params, transformed, includes_fixed, cov_type, cov_kwds, method, maxiter, full_output, disp, callback, return_params, optim_score, optim_complex_step, optim_hessian, flags, low_memory, **kwargs)
    726 else:
    727     func = self.smooth
--> 728 res = func(mlefit.params, transformed=False, includes_fixed=False,
    729            cov_type=cov_type, cov_kwds=cov_kwds)
    731 res.mlefit = mlefit
    732 res.mle_retvals = mlefit.mle_retvals

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\statespace\mlemodel.py:889, in MLEModel.smooth(self, params, transformed, includes_fixed, complex_step, cov_type, cov_kwds, return_ssm, results_class, results_wrapper_class, **kwargs)
    886 result = self.ssm.smooth(complex_step=complex_step, **kwargs)
    888 # Wrap in a results object
--> 889 return self._wrap_results(params, result, return_ssm, cov_type,
    890                           cov_kwds, results_class,
    891                           results_wrapper_class)

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\statespace\mlemodel.py:788, in MLEModel._wrap_results(self, params, result, return_raw, cov_type, cov_kwds, results_class, wrapper_class)
    785     if wrapper_class is None:
    786         wrapper_class = self._res_classes['fit'][1]
--> 788     res = results_class(self, params, result, **result_kwargs)
    789     result = wrapper_class(res)
    790 return result

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:1806, in SARIMAXResults.__init__(self, model, params, filter_results, cov_type, **kwargs)
   1804 def __init__(self, model, params, filter_results, cov_type=None,
   1805              **kwargs):
-> 1806     super(SARIMAXResults, self).__init__(model, params, filter_results,
   1807                                          cov_type, **kwargs)
   1809     self.df_resid = np.inf  # attribute required for wald tests
   1811     # Save _init_kwds

File C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\statespace\mlemodel.py:2415, in MLEResults.__init__(self, model, params, results, cov_type, cov_kwds, **kwargs)
   2412     extended_index = self.model._get_index_with_final_state()
   2413     tmp = np.transpose(self.predicted_state_cov, (2, 0, 1))
   2414     self._states.predicted_cov = pd.DataFrame(
-> 2415         np.reshape(tmp, (tmp.shape[0] * tmp.shape[1], tmp.shape[2])),
   2416         index=pd.MultiIndex.from_product(
   2417             [extended_index, columns]).swaplevel(),
   2418         columns=columns)
   2419 else:
   2420     self._states.predicted_cov = np.transpose(
   2421         self.predicted_state_cov, (2, 0, 1))

File <__array_function__ internals>:5, in reshape(*args, **kwargs)

File C:\ProgramData\Anaconda3\lib\site-packages\numpy\core\fromnumeric.py:298, in reshape(a, newshape, order)
    198 @array_function_dispatch(_reshape_dispatcher)
    199 def reshape(a, newshape, order='C'):
    200     """
    201     Gives a new shape to an array without changing its data.
    202 
   (...)
    296            [5, 6]])
    297     """
--> 298     return _wrapfunc(a, 'reshape', newshape, order=order)

File C:\ProgramData\Anaconda3\lib\site-packages\numpy\core\fromnumeric.py:57, in _wrapfunc(obj, method, *args, **kwds)
     54     return _wrapit(obj, method, *args, **kwds)
     56 try:
---> 57     return bound(*args, **kwds)
     58 except TypeError:
     59     # A TypeError occurs if the object does have such a method in its
     60     # class, but its signature is not identical to that of NumPy's. This
   (...)
     64     # Call _wrapit from within the except clause to ensure a potential
     65     # exception has a traceback chain.
     66     return _wrapit(obj, method, *args, **kwds)

KeyboardInterrupt: 

pd.DataFrame([{"period": n_seasonal_periods_range[i], "score": score} for i, score in enumerate(scores)]).sort_values("score").set_index("period").head(5)

	score
period
14	0.601745
18	0.602074
15	0.605919
32	0.609752
16	0.610036

• The best performing seasonal periods are 14 and 18.

tsa_scores = {}
tsa_scores["gas"] = evaluation_descriptive(gas_consumption_series, arima, {"order": (0, 1, 1), "seasonal_order": (0, 0, 1, 14)}, gas_pt, return_scores=True)
# arima_scores["gas"].index[-1] = "base"

AIC:  886.5067709299888

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  2.3482158739390453
MAE (train):  2.485459592488665
RMSE (test):  3.0278454704894426
RMSE (train):  3.704155770218337
AR Coefficients:  [1.]
MA Coefficients:  [ 1.         -0.53415466]
SAR Coefficients:  [1.]
SMA Coefficients:  [ 1.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.        -0.0610321]

from pmdarima.arima import auto_arima
auto_mod = auto_arima(gas_consumption_series, max_q=10, max_Q=10, m=14, method='lbfgs', with_intercept=True, seasonal=True, information_criterion="aic", random=True, n_fits=30, max_order=20, maxiter=200, trace=True, n_jobs=-1)
# res = mod.fit(disp=False) end_p=4, start_q=0, end_q=4, start_P=0, end_P=3, start_Q=0, end_Q=3,
print(auto_mod.summary())

Performing stepwise search to minimize aic
 ARIMA(2,1,2)(1,0,1)[14] intercept   : AIC=864.689, Time=1.08 sec
 ARIMA(0,1,0)(0,0,0)[14] intercept   : AIC=950.962, Time=0.07 sec
 ARIMA(1,1,0)(1,0,0)[14] intercept   : AIC=912.402, Time=0.13 sec
 ARIMA(0,1,1)(0,0,1)[14] intercept   : AIC=888.325, Time=0.21 sec
 ARIMA(0,1,0)(0,0,0)[14]             : AIC=948.990, Time=0.04 sec
 ARIMA(2,1,2)(0,0,1)[14] intercept   : AIC=862.837, Time=0.80 sec
 ARIMA(2,1,2)(0,0,0)[14] intercept   : AIC=862.603, Time=0.43 sec
 ARIMA(2,1,2)(1,0,0)[14] intercept   : AIC=862.740, Time=0.68 sec
 ARIMA(1,1,2)(0,0,0)[14] intercept   : AIC=861.575, Time=0.20 sec
 ARIMA(1,1,2)(1,0,0)[14] intercept   : AIC=861.572, Time=0.48 sec
 ARIMA(1,1,2)(2,0,0)[14] intercept   : AIC=863.403, Time=1.23 sec
 ARIMA(1,1,2)(1,0,1)[14] intercept   : AIC=863.510, Time=1.13 sec
 ARIMA(1,1,2)(0,0,1)[14] intercept   : AIC=861.684, Time=0.65 sec
 ARIMA(1,1,2)(2,0,1)[14] intercept   : AIC=864.164, Time=3.09 sec
 ARIMA(0,1,2)(1,0,0)[14] intercept   : AIC=870.390, Time=0.25 sec
 ARIMA(1,1,1)(1,0,0)[14] intercept   : AIC=860.181, Time=0.29 sec
 ARIMA(1,1,1)(0,0,0)[14] intercept   : AIC=860.683, Time=0.23 sec
 ARIMA(1,1,1)(2,0,0)[14] intercept   : AIC=862.013, Time=1.03 sec
 ARIMA(1,1,1)(1,0,1)[14] intercept   : AIC=862.117, Time=0.63 sec
 ARIMA(1,1,1)(0,0,1)[14] intercept   : AIC=860.322, Time=0.35 sec
 ARIMA(1,1,1)(2,0,1)[14] intercept   : AIC=862.699, Time=2.42 sec
 ARIMA(0,1,1)(1,0,0)[14] intercept   : AIC=888.206, Time=0.15 sec
 ARIMA(2,1,1)(1,0,0)[14] intercept   : AIC=861.478, Time=0.89 sec
 ARIMA(0,1,0)(1,0,0)[14] intercept   : AIC=949.363, Time=0.10 sec
 ARIMA(2,1,0)(1,0,0)[14] intercept   : AIC=902.393, Time=0.20 sec
 ARIMA(1,1,1)(1,0,0)[14]             : AIC=858.792, Time=0.15 sec
 ARIMA(1,1,1)(0,0,0)[14]             : AIC=859.243, Time=0.06 sec
 ARIMA(1,1,1)(2,0,0)[14]             : AIC=860.612, Time=0.29 sec
 ARIMA(1,1,1)(1,0,1)[14]             : AIC=860.724, Time=0.27 sec
 ARIMA(1,1,1)(0,0,1)[14]             : AIC=858.933, Time=0.16 sec
 ARIMA(1,1,1)(2,0,1)[14]             : AIC=861.235, Time=0.75 sec
 ARIMA(0,1,1)(1,0,0)[14]             : AIC=886.389, Time=0.09 sec
 ARIMA(1,1,0)(1,0,0)[14]             : AIC=910.471, Time=0.09 sec
 ARIMA(2,1,1)(1,0,0)[14]             : AIC=860.112, Time=0.28 sec
 ARIMA(1,1,2)(1,0,0)[14]             : AIC=860.205, Time=0.20 sec
 ARIMA(0,1,0)(1,0,0)[14]             : AIC=947.398, Time=0.06 sec
 ARIMA(0,1,2)(1,0,0)[14]             : AIC=868.816, Time=0.13 sec
 ARIMA(2,1,0)(1,0,0)[14]             : AIC=900.494, Time=0.14 sec
 ARIMA(2,1,2)(1,0,0)[14]             : AIC=861.366, Time=0.30 sec

Best model:  ARIMA(1,1,1)(1,0,0)[14]          
Total fit time: 19.758 seconds
                                      SARIMAX Results                                      
===========================================================================================
Dep. Variable:                                   y   No. Observations:                  397
Model:             SARIMAX(1, 1, 1)x(1, 0, [], 14)   Log Likelihood                -425.396
Date:                             Thu, 10 Aug 2023   AIC                            858.792
Time:                                     13:38:08   BIC                            874.717
Sample:                                 01-01-1990   HQIC                           865.101
                                      - 01-01-2023                                         
Covariance Type:                               opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1          0.4268      0.039     10.876      0.000       0.350       0.504
ma.L1         -0.8930      0.034    -26.620      0.000      -0.959      -0.827
ar.S.L14      -0.0790      0.056     -1.407      0.159      -0.189       0.031
sigma2         0.5007      0.019     26.531      0.000       0.464       0.538
===================================================================================
Ljung-Box (L1) (Q):                   0.14   Jarque-Bera (JB):              1652.41
Prob(Q):                              0.71   Prob(JB):                         0.00
Heteroskedasticity (H):               1.97   Skew:                             0.29
Prob(H) (two-sided):                  0.00   Kurtosis:                        12.99
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

tsa_scores["gas"] = tsa_scores["gas"].append(evaluation_descriptive(gas_consumption_series, arima, {"order": (1, 1, 1), "seasonal_order": (1, 0, 0, 14)}, gas_pt, return_scores=True))
# arima_scores["gas"].index[-1] = "auto_arima"

AIC:  858.7918310627217

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  2.3719863817062663
MAE (train):  2.3988763178552803
RMSE (test):  3.0323692611385846
RMSE (train):  3.5749514923420502
AR Coefficients:  [ 1.         -0.42677443]
MA Coefficients:  [ 1.         -0.89298294]
SAR Coefficients:  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.07903627]
SMA Coefficients:  [1.]

AIC and RMSE measures do not always agree on the best model parameters. The focus of the time series analysis task is to maximise accuracy which AIC does not do, thus Auto ARIMA may not be the best method.

Electricity Consumption Series

plot_acf_pacf(electricity_consumption_series, 1, 1)

Observations:

• Sharp cut off in at lag-2 in both ACF and PACF plots thus I will add both an $AR$ and $MA$ term.

# plot the residuals at lags that are multiples of 12
evaluation_descriptive(electricity_consumption_series, arima, {"order": (1, 1, 1), "seasonal_order": (0, 1, 0, 12)}, electricity_pt)

AIC:  -169.60368040488044

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  33.4891156240216
MAE (train):  22.746346433399196
RMSE (test):  44.642696955118026
RMSE (train):  32.492074822577884
AR Coefficients:  [ 1.         -0.52718763]
MA Coefficients:  [ 1.         -0.99145635]
SAR Coefficients:  [1.]
SMA Coefficients:  [1.]

Plotting ACF and PACF only at multiples of the seasonal period. (eg. 12, 24, 36, ...)

# plot the residuals at lags that are multiples of 12
evaluation_descriptive(electricity_consumption_series, arima, {"order": (1, 1, 1), "seasonal_order": (0, 1, 0, 12)}, electricity_pt, plot_seasonal=True)

AIC:  -169.60368040488044

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  33.4891156240216
MAE (train):  22.746346433399196
RMSE (test):  44.642696955118026
RMSE (train):  32.492074822577884
AR Coefficients:  [ 1.         -0.52718763]
MA Coefficients:  [ 1.         -0.99145635]
SAR Coefficients:  [1.]
SMA Coefficients:  [1.]

Observations:

• There is a more significant cut off in the ACF plot than the PACF plot. I will add either a $SMA(2)$ or $SMA(3)$ term to the model.

evaluation_descriptive(electricity_consumption_series, arima, {"order": (1, 1, 1), "seasonal_order": (0, 1, 2, 12)}, electricity_pt, plot_seasonal=True)

AIC:  -303.18467746946453

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  29.456330134631152
MAE (train):  19.834452497403294
RMSE (test):  38.18113492205217
RMSE (train):  28.992363420809962
AR Coefficients:  [ 1.         -0.52514635]
MA Coefficients:  [ 1.         -0.95647794]
SAR Coefficients:  [1.]
SMA Coefficients:  [ 1.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.69978677  0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.05553461]

More $SMA$ terms are necessary.

tsa_scores["electricity"] = evaluation_descriptive(electricity_consumption_series, arima, {"order": (1, 1, 1), "seasonal_order": (0, 1, 4, 12)}, electricity_pt, plot_seasonal=True, return_scores=True)
# arima_scores["electricity"].index[-1] = "base"

AIC:  -320.3809500915536

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  27.227980886830885
MAE (train):  19.465739099105104
RMSE (test):  34.91681622194248
RMSE (train):  28.718052172107623
AR Coefficients:  [ 1.         -0.49574718]
MA Coefficients:  [ 1.         -0.95205934]
SAR Coefficients:  [1.]
SMA Coefficients:  [ 1.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.73616726  0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.20506358  0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.16524981  0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.10777838]

from pmdarima.arima import auto_arima
auto_mod = auto_arima(electricity_consumption_series, max_q=10, max_Q=10, m=12, d=1, D=1, method='lbfgs', with_intercept=True, seasonal=True, information_criterion="aic", random=True, n_fits=30, max_order=20, maxiter=200, trace=True, n_jobs=-1)
# res = mod.fit(disp=False) end_p=4, start_q=0, end_q=4, start_P=0, end_P=3, start_Q=0, end_Q=3,
print(auto_mod.summary())

Performing stepwise search to minimize aic
 ARIMA(2,1,2)(1,1,1)[12] intercept   : AIC=-300.793, Time=8.69 sec
 ARIMA(0,1,0)(0,1,0)[12] intercept   : AIC=-79.096, Time=0.37 sec
 ARIMA(1,1,0)(1,1,0)[12] intercept   : AIC=-148.877, Time=1.25 sec
 ARIMA(0,1,1)(0,1,1)[12] intercept   : AIC=-252.102, Time=1.87 sec
 ARIMA(0,1,0)(0,1,0)[12]             : AIC=-81.049, Time=0.15 sec
 ARIMA(2,1,2)(0,1,1)[12] intercept   : AIC=-304.183, Time=7.22 sec
 ARIMA(2,1,2)(0,1,0)[12] intercept   : AIC=inf, Time=2.65 sec
 ARIMA(2,1,2)(0,1,2)[12] intercept   : AIC=inf, Time=10.46 sec
 ARIMA(2,1,2)(1,1,0)[12] intercept   : AIC=inf, Time=5.16 sec
 ARIMA(2,1,2)(1,1,2)[12] intercept   : AIC=-303.878, Time=18.35 sec
 ARIMA(1,1,2)(0,1,1)[12] intercept   : AIC=-307.153, Time=5.82 sec
 ARIMA(1,1,2)(0,1,0)[12] intercept   : AIC=inf, Time=2.21 sec
 ARIMA(1,1,2)(1,1,1)[12] intercept   : AIC=-305.753, Time=8.02 sec
 ARIMA(1,1,2)(0,1,2)[12] intercept   : AIC=inf, Time=12.45 sec
 ARIMA(1,1,2)(1,1,0)[12] intercept   : AIC=inf, Time=5.83 sec
 ARIMA(1,1,2)(1,1,2)[12] intercept   : AIC=-308.042, Time=16.87 sec
 ARIMA(1,1,2)(2,1,2)[12] intercept   : AIC=-314.640, Time=16.85 sec
 ARIMA(1,1,2)(2,1,1)[12] intercept   : AIC=-317.659, Time=34.92 sec
 ARIMA(1,1,2)(2,1,0)[12] intercept   : AIC=inf, Time=15.30 sec
 ARIMA(0,1,2)(2,1,1)[12] intercept   : AIC=-310.521, Time=10.07 sec
 ARIMA(1,1,1)(2,1,1)[12] intercept   : AIC=-307.455, Time=8.25 sec
 ARIMA(2,1,2)(2,1,1)[12] intercept   : AIC=-315.356, Time=24.35 sec
 ARIMA(1,1,3)(2,1,1)[12] intercept   : AIC=-311.584, Time=19.22 sec
 ARIMA(0,1,1)(2,1,1)[12] intercept   : AIC=-264.368, Time=5.56 sec
 ARIMA(0,1,3)(2,1,1)[12] intercept   : AIC=-315.295, Time=13.96 sec
 ARIMA(2,1,1)(2,1,1)[12] intercept   : AIC=-318.133, Time=14.97 sec
 ARIMA(2,1,1)(1,1,1)[12] intercept   : AIC=-304.852, Time=7.59 sec
 ARIMA(2,1,1)(2,1,0)[12] intercept   : AIC=inf, Time=13.38 sec
 ARIMA(2,1,1)(2,1,2)[12] intercept   : AIC=-318.674, Time=21.34 sec
 ARIMA(2,1,1)(1,1,2)[12] intercept   : AIC=-307.059, Time=20.79 sec
 ARIMA(2,1,1)(2,1,3)[12] intercept   : AIC=-312.830, Time=41.42 sec
 ARIMA(2,1,1)(1,1,3)[12] intercept   : AIC=-316.538, Time=44.20 sec
 ARIMA(1,1,1)(2,1,2)[12] intercept   : AIC=-319.608, Time=11.89 sec
 ARIMA(1,1,1)(1,1,2)[12] intercept   : AIC=-307.850, Time=14.56 sec
 ARIMA(1,1,1)(2,1,3)[12] intercept   : AIC=-318.702, Time=34.03 sec
 ARIMA(1,1,1)(1,1,1)[12] intercept   : AIC=-305.709, Time=7.57 sec
 ARIMA(1,1,1)(1,1,3)[12] intercept   : AIC=-319.887, Time=53.56 sec
 ARIMA(1,1,1)(0,1,3)[12] intercept   : AIC=-318.225, Time=33.85 sec
 ARIMA(1,1,1)(1,1,4)[12] intercept   : AIC=-318.300, Time=50.01 sec
 ARIMA(1,1,1)(0,1,2)[12] intercept   : AIC=inf, Time=9.76 sec
 ARIMA(1,1,1)(0,1,4)[12] intercept   : AIC=-320.662, Time=52.96 sec
 ARIMA(1,1,1)(0,1,5)[12] intercept   : AIC=-318.651, Time=87.65 sec
 ARIMA(1,1,1)(1,1,5)[12] intercept   : AIC=-315.600, Time=115.67 sec
 ARIMA(0,1,1)(0,1,4)[12] intercept   : AIC=-273.844, Time=24.75 sec
 ARIMA(1,1,0)(0,1,4)[12] intercept   : AIC=-252.702, Time=17.31 sec
 ARIMA(2,1,1)(0,1,4)[12] intercept   : AIC=-314.706, Time=65.43 sec
 ARIMA(1,1,2)(0,1,4)[12] intercept   : AIC=-319.284, Time=63.27 sec
 ARIMA(0,1,0)(0,1,4)[12] intercept   : AIC=-235.944, Time=16.11 sec
 ARIMA(0,1,2)(0,1,4)[12] intercept   : AIC=-310.789, Time=71.81 sec
 ARIMA(2,1,0)(0,1,4)[12] intercept   : AIC=-276.697, Time=21.32 sec
 ARIMA(2,1,2)(0,1,4)[12] intercept   : AIC=-316.534, Time=63.93 sec
 ARIMA(1,1,1)(0,1,4)[12]             : AIC=-320.381, Time=17.24 sec

Best model:  ARIMA(1,1,1)(0,1,4)[12] intercept
Total fit time: 1252.302 seconds
                                           SARIMAX Results                                           
=====================================================================================================
Dep. Variable:                                             y   No. Observations:                  397
Model:             SARIMAX(1, 1, 1)x(0, 1, [1, 2, 3, 4], 12)   Log Likelihood                 168.331
Date:                                       Fri, 11 Aug 2023   AIC                           -320.662
Time:                                               22:46:45   BIC                           -289.057
Sample:                                           01-01-1990   HQIC                          -308.126
                                                - 01-01-2023                                         
Covariance Type:                                         opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
intercept     -0.0002   8.95e-05     -2.043      0.041      -0.000   -7.47e-06
ar.L1          0.5080      0.046     11.059      0.000       0.418       0.598
ma.L1         -0.9668      0.017    -56.618      0.000      -1.000      -0.933
ma.S.L12      -0.7370      0.045    -16.318      0.000      -0.826      -0.648
ma.S.L24      -0.1966      0.055     -3.581      0.000      -0.304      -0.089
ma.S.L36       0.1452      0.071      2.038      0.042       0.006       0.285
ma.S.L48       0.1031      0.052      1.997      0.046       0.002       0.204
sigma2         0.0234      0.001     17.368      0.000       0.021       0.026
===================================================================================
Ljung-Box (L1) (Q):                   0.06   Jarque-Bera (JB):                28.03
Prob(Q):                              0.81   Prob(JB):                         0.00
Heteroskedasticity (H):               2.97   Skew:                             0.04
Prob(H) (two-sided):                  0.00   Kurtosis:                         4.32
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

the Auto ARIMA results agree with my manual term selection.

Water Consumption Series

plot_acf_pacf(water_consumption_series, 1, 0)

Observations:

• Lag-1 autocorrelation is negative and cut off in ACF lag-1, so possibly an $MA(1)$ or $MA(2)$ model

evaluation_descriptive(water_consumption_series, arima, {"order": (0, 1, 1)}, water_pt)

AIC:  932.086941951505

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  106.51578990011652
MAE (train):  77.18378594048595
RMSE (test):  130.31467060265973
RMSE (train):  106.46004568340469
AR Coefficients:  [1.]
MA Coefficients:  [ 1.         -0.55807993]
SAR Coefficients:  [1.]
SMA Coefficients:  [1.]

I will add both an $AR$ and $MA$ term.

evaluation_descriptive(water_consumption_series, arima, {"order": (1, 1, 2)}, water_pt)

AIC:  907.6587347533602

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  87.29470570320926
MAE (train):  74.6029493326991
RMSE (test):  108.12251665263804
RMSE (train):  102.87341155545565
AR Coefficients:  [ 1.         -0.62613702]
MA Coefficients:  [ 1.         -1.14845261  0.19205358]
SAR Coefficients:  [1.]
SMA Coefficients:  [1.]

n_seasonal_periods_range =  range(3, 40)
scores = []
mean_sar_coef = []
mean_sma_coef = []

# validation_gas_consumption_series = gas_consumption_series.iloc[:round(len(gas_consumption_series)*0.8)]

for seasonal_period in tqdm(n_seasonal_periods_range):
    autocorr = acf(k_diff(water_consumption_series, 1, 0), nlags=seasonal_period)[-1]
    if autocorr > 0:
        P = 1
        Q = 0
    elif autocorr < 0:
        P = 0
        Q = 1
    else:
        P = 0
        Q = 0
    params = {"order": (1, 1, 2), "seasonal_order": (P, 0, Q, seasonal_period)}
    pred, val, coef = walkforward_validation(arima, params, gas_consumption_series, return_coefficients=True)
    mean_sar_coef.append(np.mean([c["sar"][-1] for c in coef]))
    mean_sma_coef.append(np.mean([c["sma"][-1] for c in coef]))
    scores.append(mean_squared_error(pred, val, squared=False))
    print(params, scores[-1])
    
print(n_seasonal_periods_range[np.argmin(scores)])
print(n_seasonal_periods_range[np.argmax(mean_sar_coef)])
print(n_seasonal_periods_range[np.argmax(mean_sma_coef)])

  0%|          | 0/36 [00:00<?, ?it/s]

{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 4)} 0.618835772867652
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 5)} 0.6240793091480292
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 6)} 0.6192490460073472
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 7)} 0.6188499602553738
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 8)} 0.6175691310469316
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 9)} 0.6224304268155019
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 10)} 0.6166222615595816
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 11)} 0.6191014127418283
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 12)} 0.6155723618197976
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 13)} 0.6402416655300803
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 14)} 0.6043501310457127
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 15)} 0.6146356749275629
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 16)} 0.6120201650374031
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 17)} 0.6195023239685038
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 18)} 0.6045434371043298
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 19)} 0.6210884336103647
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 20)} 0.617968537336805
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 21)} 0.6194623757145359
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 22)} 0.6220287779840804
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 23)} 0.6231819944477821
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 24)} 0.6117726692394156
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 25)} 0.6204345931302986
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 26)} 0.6200359446870999
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 27)} 0.6202175290849645
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 28)} 0.6184700310318392
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 29)} 0.6204350735083098
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 30)} 0.6200839668184305
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 31)} 0.6173066670180999
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 32)} 0.6283427679948789
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 33)} 0.6226310034274388
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 34)} 0.6269106959752181
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 35)} 0.5983077226321625
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 36)} 0.6127800167885262
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 37)} 0.6756179729177871
{'order': (1, 1, 2), 'seasonal_order': (1, 0, 0, 38)} 0.6139864482608789
{'order': (1, 1, 2), 'seasonal_order': (0, 0, 1, 39)} 0.6102544282661625
35
4
6

pd.DataFrame([{"period": n_seasonal_periods_range[i], "score": score} for i, score in enumerate(scores)]).sort_values("score").set_index("period").head(5)

	score
period
35	0.598308
14	0.604350
18	0.604543
39	0.610254
24	0.611773

Observations:

• The best performing seasonal periods are 35, 14, and 18.

tsa_scores["water"] = evaluation_descriptive(water_consumption_series, arima, {"order": (1, 1, 2), "seasonal_order": (0, 0, 1, 35)}, water_pt, return_scores=True)
# arima_scores["water"].index[-1] = "base"

AIC:  909.0131294945818

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  86.98771581519591
MAE (train):  74.45884464101294
RMSE (test):  107.62458729947062
RMSE (train):  102.81327087160079
AR Coefficients:  [ 1.         -0.62926757]
MA Coefficients:  [ 1.         -1.15259135  0.19697747]
SAR Coefficients:  [1.]
SMA Coefficients:  [ 1.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.         -0.04065716]

• Including a seasonal term doesn't seem to improve performance by much.

from pmdarima.arima import auto_arima
auto_mod = auto_arima(water_consumption_series, max_q=10, max_Q=10, m=12, d=1, D=0, method='lbfgs', with_intercept=True, seasonal=True, information_criterion="aic", random=True, n_fits=30, max_order=20, maxiter=200, trace=True, n_jobs=-1)
# res = mod.fit(disp=False) end_p=4, start_q=0, end_q=4, start_P=0, end_P=3, start_Q=0, end_Q=3,
print(auto_mod.summary())

Performing stepwise search to minimize aic
 ARIMA(2,1,2)(1,0,1)[12] intercept   : AIC=909.938, Time=3.04 sec
 ARIMA(0,1,0)(0,0,0)[12] intercept   : AIC=1006.451, Time=0.11 sec
 ARIMA(1,1,0)(1,0,0)[12] intercept   : AIC=962.636, Time=0.14 sec
 ARIMA(0,1,1)(0,0,1)[12] intercept   : AIC=933.973, Time=0.12 sec
 ARIMA(0,1,0)(0,0,0)[12]             : AIC=1004.467, Time=0.04 sec
 ARIMA(2,1,2)(0,0,1)[12] intercept   : AIC=912.584, Time=1.09 sec
 ARIMA(2,1,2)(1,0,0)[12] intercept   : AIC=912.245, Time=1.09 sec
 ARIMA(2,1,2)(2,0,1)[12] intercept   : AIC=905.670, Time=6.69 sec
 ARIMA(2,1,2)(2,0,0)[12] intercept   : AIC=903.804, Time=3.20 sec
 ARIMA(1,1,2)(2,0,0)[12] intercept   : AIC=902.742, Time=1.86 sec
 ARIMA(1,1,2)(1,0,0)[12] intercept   : AIC=910.406, Time=0.60 sec
 ARIMA(1,1,2)(2,0,1)[12] intercept   : AIC=904.562, Time=3.18 sec
 ARIMA(1,1,2)(1,0,1)[12] intercept   : AIC=908.499, Time=1.93 sec
 ARIMA(0,1,2)(2,0,0)[12] intercept   : AIC=912.927, Time=0.54 sec
 ARIMA(1,1,1)(2,0,0)[12] intercept   : AIC=904.509, Time=0.84 sec
 ARIMA(1,1,3)(2,0,0)[12] intercept   : AIC=903.843, Time=2.81 sec
 ARIMA(0,1,1)(2,0,0)[12] intercept   : AIC=924.142, Time=0.40 sec
 ARIMA(0,1,3)(2,0,0)[12] intercept   : AIC=909.909, Time=0.65 sec
 ARIMA(2,1,1)(2,0,0)[12] intercept   : AIC=903.589, Time=1.10 sec
 ARIMA(2,1,3)(2,0,0)[12] intercept   : AIC=906.722, Time=2.36 sec
 ARIMA(1,1,2)(2,0,0)[12]             : AIC=900.767, Time=0.49 sec
 ARIMA(1,1,2)(1,0,0)[12]             : AIC=908.431, Time=0.22 sec
 ARIMA(1,1,2)(2,0,1)[12]             : AIC=902.588, Time=0.79 sec
 ARIMA(1,1,2)(1,0,1)[12]             : AIC=906.526, Time=0.41 sec
 ARIMA(0,1,2)(2,0,0)[12]             : AIC=910.992, Time=0.24 sec
 ARIMA(1,1,1)(2,0,0)[12]             : AIC=902.538, Time=0.30 sec
 ARIMA(2,1,2)(2,0,0)[12]             : AIC=901.825, Time=0.89 sec
 ARIMA(1,1,3)(2,0,0)[12]             : AIC=901.865, Time=0.65 sec
 ARIMA(0,1,1)(2,0,0)[12]             : AIC=922.208, Time=0.15 sec
 ARIMA(0,1,3)(2,0,0)[12]             : AIC=907.956, Time=0.41 sec
 ARIMA(2,1,1)(2,0,0)[12]             : AIC=901.614, Time=0.39 sec
 ARIMA(2,1,3)(2,0,0)[12]             : AIC=903.811, Time=2.12 sec

Best model:  ARIMA(1,1,2)(2,0,0)[12]          
Total fit time: 38.876 seconds
                                      SARIMAX Results                                      
===========================================================================================
Dep. Variable:                                   y   No. Observations:                  397
Model:             SARIMAX(1, 1, 2)x(2, 0, [], 12)   Log Likelihood                -444.383
Date:                             Thu, 10 Aug 2023   AIC                            900.767
Time:                                     13:53:00   BIC                            924.655
Sample:                                 01-01-1990   HQIC                           910.231
                                      - 01-01-2023                                         
Covariance Type:                               opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1          0.6724      0.089      7.514      0.000       0.497       0.848
ma.L1         -1.2131      0.100    -12.130      0.000      -1.409      -1.017
ma.L2          0.2446      0.085      2.865      0.004       0.077       0.412
ar.S.L12       0.0547      0.051      1.081      0.280      -0.044       0.154
ar.S.L24       0.1594      0.061      2.627      0.009       0.040       0.278
sigma2         0.5499      0.028     19.819      0.000       0.496       0.604
===================================================================================
Ljung-Box (L1) (Q):                   0.01   Jarque-Bera (JB):               110.08
Prob(Q):                              0.90   Prob(JB):                         0.00
Heteroskedasticity (H):               1.56   Skew:                            -0.27
Prob(H) (two-sided):                  0.01   Kurtosis:                         5.53
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

tsa_scores["water"] = tsa_scores["water"].append(evaluation_descriptive(water_consumption_series, arima, {"order": (1, 1, 2), "seasonal_order": (2, 0, 0, 12)}, water_pt, return_scores=True))
# arima_scores["water"].index[-1] = "auto_arima"

AIC:  900.7667295535439

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  82.21341311552578
MAE (train):  73.32313321518177
RMSE (test):  102.17688106222886
RMSE (train):  101.6572975288717
AR Coefficients:  [ 1.         -0.67243027]
MA Coefficients:  [ 1.         -1.21312822  0.24459481]
SAR Coefficients:  [ 1.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.05470304  0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
 -0.15940979]
SMA Coefficients:  [1.]

def plot_model_params(series, model, params, transformer, title="", past_length=36, test_length=12, forecast_length=12, is_hwes=False, plot_conf_int=True, **kwargs):
    if not is_hwes:
        forecast_wrapper = model(series[:-test_length], params).fit().get_prediction(start=len(series)-past_length, end=len(series)+forecast_length)
        forecasted_series = reverse_transform_series(forecast_wrapper.predicted_mean, transformer)
        if plot_conf_int:
            lower_conf_int = reverse_transform_series(forecast_wrapper.conf_int().loc[series.index[-test_length]:, "lower y"], transformer)
            upper_conf_int = reverse_transform_series(forecast_wrapper.conf_int().loc[series.index[-test_length]:, "upper y"], transformer)
    else: 
        forecasted_series = reverse_transform_series(model(series[:-test_length], params).fit().predict(start=len(series)-past_length, end=len(series)+forecast_length), transformer)
    # validate_plot(gas_consumption_series, arima, , gas_pt)
    # {'order': (3, 1, 0), 'seasonal_order': (3, 0, 2, 24)}
    # (validation, transformer)
    # forecasted_series = first_electricity_consumption_series.append(forecast_wrapper.predicted_mean).cumsum()
    
    
    sns.set(style="darkgrid")
    plt.figure(**kwargs)
    sns.lineplot(data=reverse_transform_series(series, transformer)[-past_length:], linestyle='solid', label="Train")
    sns.lineplot(data=forecasted_series, linestyle='solid', alpha=0.7, label="Predicted")

    test = reverse_transform_series(series, transformer)[-test_length: ]

    predict = forecasted_series.loc[test.index[0]: test.index[-1]]
    
    print(f"MAE (test): {mean_absolute_error(test, predict)}")
    print(f"RMSE (test): {mean_squared_error(test, predict, squared=False)}")

    sns.lineplot(data=test, linestyle='solid', label="Test")
    
    if plot_conf_int:
        plt.fill_between(lower_conf_int.index, lower_conf_int, upper_conf_int, alpha=0.3, label="Confidence Interval")
        
    if title != "":
        plt.title(title)
    plt.xlabel('Date')
    plt.ylabel('Value')
    # plt.legend(['Actual', 'Confidence Interval', 'Predicted'])

    plt.axvline(x=series.index[-test_length], color='k', linestyle='--', alpha=0.1)
    plt.axvline(x=series.index[-1], color='k', linestyle='--', alpha=0.1)
    plt.legend(loc="upper left")
    plt.show()

Initial Tuning Results of SARIMA

plot_model_params(
    gas_consumption_series, 
    arima, 
    {"order": (0, 1, 2), "seasonal_order": (1, 0, 0, 14)}, 
    gas_pt, 
    title='Gas Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

plot_model_params(
    electricity_consumption_series, 
    arima, 
    {"order": (1, 1, 1), "seasonal_order": (0, 1, 4, 12)}, 
    electricity_pt, 
    title='Electricity Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

plot_model_params(
    water_consumption_series, 
    arima, 
    {"order": (1, 1, 2), "seasonal_order": (2, 0, 0, 12)}, 
    water_pt, 
    title='Water Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

MAE (test): 2.094432522117321
RMSE (test): 2.4842377523333896

MAE (test): 27.7254377559711
RMSE (test): 38.56942852276296

MAE (test): 82.82420980989968
RMSE (test): 105.59883699989969

VARMAX

Since we rejected the null hypothesis in Granger's Causality test when testing for some combination of the variables, namely, Gas Consumption causing Water Consumption and Electricity Consumption causing Gas Consumption, I will use VARMAX which is a form of multivariate time series model. It can take multiple variables as a Vector and apply many of the same Auto Regressive and Moving Average techniques to it. This could result in better out-of-sample prediction results since more data is available.

def varmax_rmse_wf_validation_gas(model, params, series, transformer):
    varmax_validation, varmax_prediction = walkforward_validation(varmax_, params, varmax_no_na_df, flatten=False, gap=2, step=6, verbose=False)

    for i in range(len(varmax_validation)):
        varmax_validation[i] = k_undiff(varmax_validation[i], [varmax_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
    for i in range(len(varmax_prediction)):
        varmax_prediction[i] = k_undiff(varmax_prediction[i], [varmax_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]

        validations = varmax_validation[0]
    forecasts = varmax_prediction[0]
    for i in range(1, len(varmax_prediction)):
        validations = validations.append(varmax_validation[i])
        forecasts = forecasts.append(varmax_prediction[i])
    
    return -mean_squared_error(reverse_transform_series(validations.iloc[:, 0], transformer), reverse_transform_series(forecasts.iloc[:, 0], transformer), squared=False)

def varmax_rmse_wf_validation_water(model, params, series, transformer):
    varmax_validation, varmax_prediction = walkforward_validation(varmax_, params, varmax_no_na_df, flatten=False, gap=2, step=6, verbose=False)

    for i in range(len(varmax_validation)):
        varmax_validation[i] = k_undiff(varmax_validation[i], [varmax_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
    for i in range(len(varmax_prediction)):
        varmax_prediction[i] = k_undiff(varmax_prediction[i], [varmax_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]

        validations = varmax_validation[0]
    forecasts = varmax_prediction[0]
    for i in range(1, len(varmax_prediction)):
        validations = validations.append(varmax_validation[i])
        forecasts = forecasts.append(varmax_prediction[i])
    
    return -mean_squared_error(reverse_transform_series(validations.iloc[:, 1], transformer), reverse_transform_series(forecasts.iloc[:, 1], transformer), squared=False)

def varmax_evaluation_descriptive(df, ori_df, model, params, idx, transformer, return_sets=False):
    varmax_validation, varmax_prediction, varmax_train, varmax_train_pred = walkforward_validation(model, params, df, flatten=False, gap=2, step=2, verbose=False, return_train=True)
    
    for i in range(len(varmax_validation)):
        varmax_validation[i] = k_undiff(varmax_validation[i], [ori_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
    for i in range(len(varmax_prediction)):
        varmax_prediction[i] = k_undiff(varmax_prediction[i], [ori_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
    for i in range(len(varmax_train)):
        varmax_train[i] = k_undiff(varmax_train[i], [ori_df.loc[varmax_train[i].index[0] - pd.DateOffset(months=1):varmax_train[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
    for i in range(len(varmax_train_pred)):
        varmax_train_pred[i] = k_undiff(varmax_train_pred[i], [ori_df.loc[varmax_train_pred[i].index[0] - pd.DateOffset(months=1):varmax_train_pred[i].index[0] - pd.DateOffset(months=1)]], None)[1:]

    validations = varmax_validation[0]
    forecasts = varmax_prediction[0]
    trains = varmax_train[0]
    train_preds = varmax_train_pred[0]
    for i in range(1, len(varmax_prediction)):
        validations = validations.append(varmax_validation[i])
        forecasts = forecasts.append(varmax_prediction[i])
        trains = trains.append(varmax_train[i])
        train_preds = train_preds.append(varmax_train_pred[i])
        
    scores = pd.DataFrame()
    
    scores["mae_test"] = [mean_absolute_error(reverse_transform_series(validations.iloc[:, idx], transformer), reverse_transform_series(forecasts.iloc[:, idx], transformer))]
    scores["mae_train"] = [mean_absolute_error(reverse_transform_series(trains.iloc[:, idx], transformer), reverse_transform_series(train_preds.iloc[:, idx], transformer))]
    scores["rmse_test"] = [mean_squared_error(reverse_transform_series(validations.iloc[:, idx], transformer), reverse_transform_series(forecasts.iloc[:, idx], transformer), squared=False)]
    scores["rmse_train"] = [mean_squared_error(reverse_transform_series(trains.iloc[:, idx], transformer), reverse_transform_series(train_preds.iloc[:, idx], transformer), squared=False)]
    
    print("MAE (test): ", scores["mae_test"][0])
    print("MAE (train): ", scores["mae_train"][0])
    print("RMSE (test): ", scores["rmse_test"][0])
    print("RMSE (train): ", scores["rmse_train"][0])
    
    return scores

varmax_df = df.copy()

varmax_df["Gas Consumption (tons)"] = gas_consumption_series
varmax_df["Water Consumption (tons)"] = water_consumption_series
varmax_df["Electricity Consumption (MWh)"] = electricity_consumption_series

varmax_gas_water_df = varmax_df.drop(columns=["Electricity Consumption (MWh)"])
varmax_elec_gas_df = varmax_df.drop(columns=["Water Consumption (tons)"])

varmax_gas_water_diffed_df = varmax_gas_water_df.copy()
varmax_elec_gas_diffed_df = varmax_elec_gas_df.copy()

varmax_gas_water_diffed_df["Gas Consumption (tons)"], _, _ = k_diff(varmax_df["Gas Consumption (tons)"], 1, 0, return_discards=True)
varmax_gas_water_diffed_df["Water Consumption (tons)"], _, _ = k_diff(varmax_df["Water Consumption (tons)"], 1, 0, return_discards=True)
varmax_elec_gas_diffed_df["Gas Consumption (tons)"], _, _ = k_diff(varmax_df["Gas Consumption (tons)"], 1, 0, return_discards=True)
varmax_elec_gas_diffed_df["Electricity Consumption (MWh)"], _, _ = k_diff(varmax_df["Electricity Consumption (MWh)"], 1, 1, return_discards=True)

varmax_gas_water_diffed_df = varmax_gas_water_diffed_df.dropna()
varmax_elec_gas_diffed_df = varmax_elec_gas_diffed_df.dropna()

varmax_gas_water_diffed_df

	Gas Consumption (tons)	Water Consumption (tons)
DATE
1990-02-01	-0.482310	0.694593
1990-03-01	0.330299	-0.978140
1990-04-01	0.345597	0.031546
1990-05-01	0.652248	-0.201848
1990-06-01	0.287893	0.110997
...	...	...
2022-09-01	0.098628	-0.375552
2022-10-01	0.794732	-1.046254
2022-11-01	-0.153205	-0.215973
2022-12-01	0.267551	0.032495
2023-01-01	-0.209998	-0.646975

396 rows × 2 columns

varmax_elec_gas_diffed_df

	Gas Consumption (tons)	Electricity Consumption (MWh)
DATE
1991-02-01	-1.112116	-0.222739
1991-03-01	-0.086600	0.156794
1991-04-01	-0.021716	-0.013145
1991-05-01	0.021716	0.058557
1991-06-01	0.194272	0.083537
...	...	...
2022-09-01	0.098628	0.164666
2022-10-01	0.794732	0.410814
2022-11-01	-0.153205	0.152864
2022-12-01	0.267551	-0.161986
2023-01-01	-0.209998	0.841394

384 rows × 2 columns

p_range = range(0, 5)
q_range = range(0, 5)

# Gas cause Water

varmax_scores = []
varmax_params = []

for p in p_range:
    for q in q_range:
        if not(p == 0 and q == 0):
            print(p, q)
            varmax_params.append((p, q))
            varmax_validation, varmax_prediction = walkforward_validation(varmax, {"order": (p, q)}, varmax_gas_water_diffed_df, flatten=False, step=4)

            for i in range(len(varmax_validation)):
                varmax_validation[i] = k_undiff(varmax_validation[i], [varmax_gas_water_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
            for i in range(len(varmax_prediction)):
                varmax_prediction[i] = k_undiff(varmax_prediction[i], [varmax_gas_water_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]

                validations = varmax_validation[0]
            forecasts = varmax_prediction[0]
            for i in range(1, len(varmax_prediction)):
                validations = validations.append(varmax_validation[i])
                forecasts = forecasts.append(varmax_prediction[i])

            varmax_score = mean_squared_error(reverse_transform_series(validations.iloc[:, 1], water_pt), reverse_transform_series(forecasts.iloc[:, 1], water_pt), squared=False)
            print("Water Consumption (VARMAX) RMSE", varmax_score)
            varmax_scores.append(varmax_score)

print("Best Score", min(varmax_scores))
print("Best parameters", varmax_params[np.argmin(varmax_scores)])

0 1

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 138.43052066224047
0 2

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 130.00456959101652
0 3

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 121.1440261251107
0 4

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 120.80213269937707
1 0

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 154.6924280134657
1 1

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 119.72439640125282
1 2

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 123.55503774834051
1 3

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 122.03704136069426
1 4

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 120.34172372783749
2 0

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 151.05024180685731
2 1

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 113.3434546992668
2 2

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 121.52509091555376
2 3

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 120.57761186131353
2 4

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 118.27485418202397
3 0

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 148.17448519075833
3 1

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 120.42028424340053
3 2

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 119.5027571935542
3 3

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 120.57162619132502
3 4

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 118.65532343899592
4 0

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 143.74232717088847
4 1

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 116.6033154346191
4 2

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 122.67214282230591
4 3

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 121.53547020523655
4 4

  0%|          | 0/7 [00:00<?, ?it/s]

Water Consumption (VARMAX) RMSE 121.92980329386958
Best Score 113.3434546992668
Best parameters (2, 1)

Optimal p, q for VARMA (Gas causing Water): 2, 1

# Electricity cause Gas

varmax_scores = []
varmax_params = []

for p in p_range:
    for q in q_range:
        if not(p == 0 and q == 0):
            print(p, q)
            varmax_params.append((p, q))
            varmax_validation, varmax_prediction = walkforward_validation(varmax, {"order": (p, q)}, varmax_elec_gas_diffed_df, flatten=False, step=4)

            for i in range(len(varmax_validation)):
                varmax_validation[i] = k_undiff(varmax_validation[i], [varmax_elec_gas_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]
            for i in range(len(varmax_prediction)):
                varmax_prediction[i] = k_undiff(varmax_prediction[i], [varmax_elec_gas_df.loc[varmax_validation[i].index[0] - pd.DateOffset(months=1):varmax_validation[i].index[0] - pd.DateOffset(months=1)]], None)[1:]

                validations = varmax_validation[0]
            forecasts = varmax_prediction[0]
            for i in range(1, len(varmax_prediction)):
                validations = validations.append(varmax_validation[i])
                forecasts = forecasts.append(varmax_prediction[i])

            varmax_score = mean_squared_error(reverse_transform_series(validations.iloc[:, 0], gas_pt), reverse_transform_series(forecasts.iloc[:, 1], gas_pt), squared=False)
            print("Gas Consumption (VARMAX) RMSE", varmax_score)
            varmax_scores.append(varmax_score)

print("Best Score", min(varmax_scores))
print("Best parameters", varmax_params[np.argmin(varmax_scores)])

0 1

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.6516909933974175
0 2

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.201843951061939
0 3

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.142292319526405
0 4

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.2107371363773782
1 0

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.9289804127635772
1 1

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.3371987081926364
1 2

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1170247086827727
1 3

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.141188135667428
1 4

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1919711511142617
2 0

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.6655398558621313
2 1

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.4382964541149184
2 2

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.205094822960973
2 3

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.180604517236145
2 4

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1616400701488288
3 0

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.6377081202190724
3 1

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.427762820462859
3 2

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.2222999165349013
3 3

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.198184407668339
3 4

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.181953167238836
4 0

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.5147468591417312
4 1

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.3156200723616327
4 2

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1774290976445942
4 3

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1890316967014263
4 4

  0%|          | 0/7 [00:00<?, ?it/s]

Gas Consumption (VARMAX) RMSE 3.1799769472442208
Best Score 3.1170247086827727
Best parameters (1, 2)

Optimal p, q for VARMA (Electricity cause Gas): 1, 2

tsa_scores["water"] = tsa_scores["water"].append(varmax_evaluation_descriptive(varmax_gas_water_diffed_df, varmax_gas_water_df, varmax, {"order": (2, 1)}, 0, water_pt))
tsa_scores["gas"] = tsa_scores["gas"].append(varmax_evaluation_descriptive(varmax_elec_gas_diffed_df, varmax_elec_gas_df, varmax, {"order": (1, 2)}, 0, gas_pt))

MAE (test):  60.9644393164263
MAE (train):  782.5407044448101
RMSE (test):  76.32959123852366
RMSE (train):  1006.8682814470981
MAE (test):  2.649907593782821
MAE (train):  14.978199307181153
RMSE (test):  3.2873323741756915
RMSE (train):  18.214567647123523

past_length = 96
forecast_length = 12
test_length = 24
params = {"order": (2, 1)}

varmax_fit = varmax(varmax_gas_water_diffed_df[:-test_length], params).fit()
in_sample_wrapper = varmax_fit.get_prediction(start=0, end=len(varmax_gas_water_diffed_df)-test_length)
forecast_wrapper = varmax_fit.get_prediction(start=len(varmax_gas_water_diffed_df)-test_length, end=len(varmax_gas_water_diffed_df)+forecast_length)

in_sample_series = k_undiff(
    in_sample_wrapper.predicted_mean, 
    [varmax_gas_water_df.loc[in_sample_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1):in_sample_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1)]], 
    None)[1:]
forecasted_series = k_undiff(
    forecast_wrapper.predicted_mean, 
    [varmax_gas_water_df.loc[forecast_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1):forecast_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1)]], 
    None)[1:]

water_in_sample_series = reverse_transform_series(in_sample_series.iloc[:, 1], water_pt)
water_forecasted_series = reverse_transform_series(forecasted_series.iloc[:, 1], water_pt)

series = gas_consumption_series

# Gas
sns.set(style="darkgrid")
plt.figure(figsize=(10, 6))

sns.lineplot(data=reverse_transform_series(water_consumption_series, water_pt)[-past_length:], linestyle='solid', label="Train")
sns.lineplot(data=water_in_sample_series[-past_length+test_length:], linestyle='solid', label="In-Sample")
sns.lineplot(data=water_forecasted_series, linestyle='solid', alpha=0.7, label="Predicted")

test = reverse_transform_series(water_consumption_series, water_pt)[-test_length: ]

predict = gas_forecasted_series.loc[test.index[0]: test.index[-1]]

print(f"RMSE: {mean_squared_error(test, predict, squared=False)}")

sns.lineplot(data=test, linestyle='solid', label="Test")
plt.title("Water Consumption VARMAX Forecast")
plt.xlabel('Date')
plt.ylabel('Value')

plt.axvline(x=series.index[-test_length], color='k', linestyle='--', alpha=0.1)
plt.axvline(x=series.index[-1], color='k', linestyle='--', alpha=0.1)
plt.legend(loc="upper left")
plt.title
plt.show()

RMSE: 473.71394805294153

past_length = 96
forecast_length = 12
test_length = 24
params = {"order": (1, 2)}

varmax_fit = varmax(varmax_elec_gas_diffed_df[:-test_length], params).fit()
in_sample_wrapper = varmax_fit.get_prediction(start=0, end=len(varmax_elec_gas_diffed_df)-test_length)
forecast_wrapper = varmax_fit.get_prediction(start=len(varmax_elec_gas_diffed_df)-test_length, end=len(varmax_elec_gas_diffed_df)+forecast_length)

in_sample_series = k_undiff(
    in_sample_wrapper.predicted_mean, 
    [varmax_elec_gas_df.loc[in_sample_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1):in_sample_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1)]], 
    None)[1:]
forecasted_series = k_undiff(
    forecast_wrapper.predicted_mean, 
    [varmax_elec_gas_df.loc[forecast_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1):forecast_wrapper.predicted_mean.iloc[:, 0].index[0] - pd.DateOffset(months=1)]], 
    None)[1:]

gas_in_sample_series = reverse_transform_series(in_sample_series.iloc[:, 0], gas_pt)
gas_forecasted_series = reverse_transform_series(forecasted_series.iloc[:, 0], gas_pt)

series = gas_consumption_series

# Gas
sns.set(style="darkgrid")
plt.figure(figsize=(10, 6))

sns.lineplot(data=reverse_transform_series(gas_consumption_series, gas_pt)[-past_length:], linestyle='solid', label="Train")
sns.lineplot(data=gas_in_sample_series[-past_length+test_length:], linestyle='solid', label="In-Sample")
sns.lineplot(data=gas_forecasted_series, linestyle='solid', alpha=0.7, label="Predicted")

test = reverse_transform_series(gas_consumption_series, gas_pt)[-test_length: ]

predict = gas_forecasted_series.loc[test.index[0]: test.index[-1]]

print(f"RMSE: {mean_squared_error(test, predict, squared=False)}")

sns.lineplot(data=test, linestyle='solid', label="Test")
plt.title("Gas Consumption VARMAX Forecast")
plt.xlabel('Date')
plt.ylabel('Value')

plt.axvline(x=series.index[-test_length], color='k', linestyle='--', alpha=0.1)
plt.axvline(x=series.index[-1], color='k', linestyle='--', alpha=0.1)
plt.legend(loc="upper left")
plt.title
plt.show()

RMSE: 3.7108300977177993

Observations

• Even though the test RMSE tells me that VARMAX performs well, the plots look extremely irregular indicating something is off with the predictions.

Hyperparameter Tuning: Artificial Bee Colony

Since there is a large number of parameters to tune for some of the models, I opt to use the ABC tuning algorithm instead of a more exhaustive method like GridSearch to reduce tuning time.

bool_option = ([True, False], "str")

all_model_settings = [
    {
        "tsa_model": hwes,
        "tunable_space": {
            "trend": (["add"], "str"),
#             "trend": (["add", "mul", None], "str"),
            "damped_trend": bool_option,
            "seasonal": (["add", None], "str"),
#             "seasonal": (["add", "mul", None], "str"),
            "seasonal_periods": (2, 48, "int"),
            "initialization_method": (["estimated", "heuristic"], "str"),
#             "use_boxcox": bool_option
        },
        "default_hyperparameters": {
            "trend": "add",
            "damped_trend": False,
            "seasonal": None,
            "seasonal_periods": None,
            "initialization_method": "estimated",
#             "use_boxcox": False
        },
        "common_parameters": {
            "use_boxcox": False
        }
    },
    {
        "tsa_model": arima_,
        "tunable_space": {
            "p": (0, 3, "int"),
#             "d": (0, 3, "int"),
            "q": (0, 3, "int"),
            "P": (0, 3, "int"),
#             "D": (0, 3, "int"),
            "Q": (0, 3, "int"),
#             "M": (12, 36, "int"),
#             "trend": (["n", "t"], "str"),
#             "trend": (["c"], "str"),
#             "trend": (["n", "c", "t", "ct"], "str"),
            "enforce_stationarity": bool_option,
            "enforce_invertibility": bool_option,
#             "concentrate_scale": bool_option,
            "trend_offset": (0, 5, "int")
        },
        "default_hyperparameters": {
            "p": 0,
#             "d": 0,
            "q": 0,
            "P": 0,
#             "D": 0,
            "Q": 0,
#             "M": 12,
#             "trend": "c",
            
            "enforce_stationarity": True,
            "enforce_invertibility": True,
#             "concentrate_scale": bool_option,
            "trend_offset": 1
        },
        "common_parameters": {
#             "n_jobs": -1,
#             "random_state": 23
        }
#         "scorer": opt_pb_f1_scorer
    },
    {
        "tsa_model": sarimax,
        "tunable_space": {
#             "p": (0, 5, "int"),
# #             "d": (0, 5, "int"),
#             "q": (0, 5, "int"),
#             "P": (0, 5, "int"),
# #             "D": (0, 5, "int"),
#             "Q": (0, 5, "int"),
#             "M": (6, 24, "int"),
#             "trend": (["n", "t"], "str"),
            "trend": (["n", "c", "t", "ct"], "str"),
            "measurement_error": bool_option,
            "simple_differencing": bool_option,
#             "hamilton_representation": bool_option,
            "enforce_stationarity": bool_option,
            "enforce_invertibility": bool_option,
#             "concentrate_scale": bool_option,
            "trend_offset": (0, 5, "int")
        },
        "default_hyperparameters": {
#             "p": 1,
# #             "d": 0,
#             "q": 0,
#             "P": 0,
# #             "D": 0,
#             "Q": 0,
#             "M": 12,
            "trend": "n",
            "measurement_error": False,
            "simple_differencing": False,
#             "hamilton_representation": False,
            "enforce_stationarity": True,
            "enforce_invertibility": True,
#             "concentrate_scale": bool_option,
            "trend_offset": 1
        },
        "common_parameters": {
#             "n_jobs": -1,
#             "random_state": 23
        }
#         "scorer": opt_pb_f1_scorer
    },
    {
        "tsa_model": varmax_,
        "tunable_space": {
            "p": (0, 4, "int"),
#             "d": (0, 5, "int"),
            "q": (0, 4, "int"),
            "error_cov_type": (['diagonal', 'unstructured'], 'str'),
            "trend": (["n", "c", "t", "ct"], "str"),
            "measurement_error": bool_option,
            "simple_differencing": bool_option,
#             "hamilton_representation": bool_option,
            "enforce_stationarity": bool_option,
            "enforce_invertibility": bool_option,
#             "concentrate_scale": bool_option,
            "trend_offset": (0, 5, "int")
        },
        "default_hyperparameters": {
            "p": 1,
# #             "d": 0,
            "q": 0,
#             "P": 0,
# #             "D": 0,
#             "Q": 0,
#             "M": 12,
            "trend": "n",
            "error_cov_type": "unstructured",
            "measurement_error": False,
            "simple_differencing": False,
#             "hamilton_representation": False,
            "enforce_stationarity": True,
            "enforce_invertibility": True,
#             "concentrate_scale": bool_option,
            "trend_offset": 1
        },
        "common_parameters": {
#             "n_jobs": -1,
#             "random_state": 23
        }
#         "scorer": opt_pb_f1_scorer
    },
]

import random
import copy
import math

class Bee:
    def __init__(self, params, score, stay_limit, employed):
        self._params = params
        self._score = score
        self._is_employed = employed
        self._stay_limit = stay_limit
        self._abandon_count = 0
        # print(f'stay limit: {self._stay_limit}')
        
    def abandon(self):
        self._abandon_count += 1
        if self._abandon_count >= self._stay_limit:
            return True
        else:
            return False
        
    def is_better_food(self, new_score):
        if new_score > self._score:
            return True
        else:
            return False

from multiprocess import Pool
from bisect import bisect

class ArtificalBeeColony:
    def __init__(self,
                 tsa_model,
                 series,
                 search_space,
                 common_params, 
                 default_params, 
                 fitness_fn, 
                 fitness_fn_params, 
                 transformer,
                 n_employed_bees, 
                 n_processes, 
#                  scoring_metric,
                 stay_limit=5, 
                 baseline_score=0, 
                 mutation_chance=0.5, 
                 mutation_std=0.1, 
                 ignore_warnings=True
                ):
        self._tsa_model = tsa_model
        self._series = series
        self._search_space = search_space
        self._common_params = common_params
        self._default_params = default_params
        self._fitness_fn = fitness_fn
        self._fitness_fn_params = fitness_fn_params
        self._transformer = transformer
#         self._scoring_metric = scoring_metric
        self._n_employed_bees = n_employed_bees
        self._best_score = 0
        self._best_params = None
        self._n_processes = n_processes
        self._mutation_chance = mutation_chance
        self._mutation_std = mutation_std
        self._stay_limit = stay_limit
        self._baseline_score = baseline_score
        self._ignore_warnings = ignore_warnings
        self._bees = []
    
    def score(self, params):
        print(params)
        params = copy.deepcopy(params)
        params.update(self._common_params)
#         tsa_model = self._tsa_model.fit(params, **self._common_params)
        if self._ignore_warnings:
            score = None
            try: 
                score = self._fitness_fn(self._tsa_model, params, self._series, self._transformer)
                return (params, score)
            except KeyboardInterrupt as err:
                raise KeyboardInterrupt(err)
            except:
                return (params, self._baseline_score)
        else:
            score = self._fitness_fn(self._tsa_model, params, self._series, self._transformer)
#             print("bruh")
            print(score)
            return (params, score)
        
#     def employed_bees_eval(self):
    
    def clamp(self, value, c_range):
#         print("clamp:", value, c_range)
        if value > c_range[1]:
            value = c_range[1]
        if value < c_range[0]:
            value = c_range[0]
#         print(value)
        return value

    def random_food(self):
        food = {}
        for key, value in self._search_space.items():
            if value[-1] == "int":
                food[key] = np.random.randint(value[0], value[1])
            elif value[-1] == "float":
                food[key] = np.random.uniform(value[0], value[1])
            elif value[-1] == "str":
                food[key] = np.random.choice(value[0])
            elif value[-1] == "arr":
                food[key] = np.random.randint(value[0], value[1], size=np.random.randint(1, value[2]))

            if value[-1] == "int" or value[-1] == "float":
                food[key] = self.clamp(food[key], value)
            if value[-1] == "arr":
                food[key] = [self.clamp(x, value) for x in food[key]]
        return food
    
    def mutate_food(self, params):
        food = {}
        for key, value in self._search_space.items():
            if random.random() < self._mutation_chance:
                if params[key] == 0:
                    mute_param = 1
                else:
                    mute_param = params[key]
                if value[-1] == "int":
                    if params[key] is None:
                        params[key] = 0
                        mute_param = 1
                    food[key] = int(params[key] + mute_param * np.random.normal(loc=0, scale=self._mutation_std))
                elif value[-1] == "float":
                    if params[key] is None:
                        params[key] = 0
                        mute_param = 1
                    food[key] = params[key] + mute_param * np.random.normal(loc=0, scale=self._mutation_std)
                elif value[-1] == "str":
#                     print(params[key])
                    food[key] = np.random.choice(value[0])
                elif value[-1] == "arr":
                    food[key] = [int(x + x * np.random.normal(loc=0, scale=self._mutation_std)) for x in params[key]]

                if value[-1] == "int" or value[-1] == "float":
                    food[key] = self.clamp(food[key], value)
                if value[-1] == "arr":
                    food[key] = [self.clamp(x, value) for x in food[key]]
            else:
                food[key] = params[key]
        return food

    def choose_bee(self):
        for b in self._bees:
            if math.isnan(b._score):
                b._score = self._baseline_score
        lowest_score = np.min([b._score for b in self._bees])
        fitness_sum = sum((b._score - lowest_score) for b in self._bees)
        if fitness_sum == 0:
            idx = np.random.randint(len(self._bees))
        else:
            probabilities = [(b._score - self._baseline_score) / fitness_sum for b in self._bees]
            cdf_vals = []
            cumsum = 0
            for p in probabilities:
                cumsum += p
                cdf_vals.append(cumsum)
            idx = bisect(cdf_vals, random.random())
        while idx >= len(self._bees):
            idx = np.random.randint(len(self._bees))
        # print(self._bees[idx] == self.best())
        return self._bees[idx]

    def best(self):
        idx = np.argmax([b._score for b in self._bees])
        # print([b._score for b in self._bees])
        return self._bees[idx]
    
    def init_colony(self):
#         print(self.score(self._default_params))
        self._bees = []

        if self._n_processes > 1:
            bee_pool = Pool(processes=self._n_processes)
        employed_results = []
            
        for _ in range(self._n_employed_bees-1):
            params = self.random_food()
            if self._n_processes > 1:
                employed_results.append(bee_pool.apply_async(
                    self.score, [params]
                ))
            else:
                employed_results.append(self.score(
                    params
                ))
                
        if self._n_processes > 1:
            bee_pool.close()
            bee_pool.join()
            employed_results = [r.get() for r in employed_results]
            
        for result in employed_results:

            self._bees.append(Bee(
                result[0],
                result[1],
                self._stay_limit,
                True
            ))
        
        
        self._bees.append(Bee(
            self._default_params,
            self.score(self._default_params)[1],
            True,
            self._stay_limit)
        )

        # print(self._bees[-1]._score)
          
        if self._n_processes > 1:
            bee_pool = Pool(processes=self._n_processes)
        onlooker_results = []

        for _ in range(self._n_employed_bees):

            chosen_employed = self.choose_bee()
            neighbor_food = self.mutate_food(
                chosen_employed._params
            )

            if self._n_processes > 1:
                onlooker_results.append(bee_pool.apply_async(
                    self.score, [params]
                ))
            else:
                onlooker_results.append(self.score(
                    params
                ))

        if self._n_processes > 1:

            bee_pool.close()
            bee_pool.join()
            onlooker_results = [r.get() for r in onlooker_results]

        for result in onlooker_results:

            self._bees.append(Bee(
                result[0],
                result[1],
                self._stay_limit,
                False
            ))
  
    def next_generation(self):
        if self._n_processes > 1:
            bee_pool = Pool(processes=self._n_processes)
        new_employed_results = []
        new_onlooker_results = []
        new_position_results = []
        current_positions = []
        best_bee = self.best()

        next_bee_generation = []

        for bee in self._bees:

            if bee == best_bee:

                next_bee_generation.append(bee)

            else:
                if bee.abandon():

                    if bee._is_employed:

                        new_params = self.random_food()
                        if self._n_processes > 1:
                            new_employed_results.append(bee_pool.apply_async(
                                self.score,
                                [new_params,]
                            ))
                        else:
                            new_employed_results.append(self.score(
                                new_params
                            ))

                    else:

                        chosen_bee = self.choose_bee()
                        neighbor_food = self.mutate_food(
                            chosen_bee._params
                        )
                        if self._n_processes > 1:
                            new_onlooker_results.append(bee_pool.apply_async(
                                self.score,
                                [neighbor_food]
                            ))
                        else:
                            new_onlooker_results.append(self.score(
                                neighbor_food
                            ))

                else:

                    current_positions.append(bee)

                    if bee._is_employed:
                        neighbor_food = self.mutate_food(bee._params)
                    else: 
                        chosen_bee = self.choose_bee()
                        neighbor_food = self.mutate_food(
                            chosen_bee._params
                        )
                    if self._n_processes > 1:
                        new_position_results.append(bee_pool.apply_async(
                            self.score,
                            [neighbor_food]
                        ))
                    else:
                        new_position_results.append(self.score(
                            neighbor_food
                        ))


        if self._n_processes > 1:

            bee_pool.close()
            bee_pool.join()
            new_employed_results = [r.get() for r in new_employed_results]
            new_onlooker_results = [r.get() for r in new_onlooker_results]
            new_position_results = [r.get() for r in new_position_results]

        for result in new_employed_results:

            next_bee_generation.append(Bee(
                result[0],
                result[1],
                self._stay_limit,
                True
            ))

        for result in new_onlooker_results:

            next_bee_generation.append(Bee(
                result[0],
                result[1],
                self._stay_limit,
                False
            ))

        for idx, result in enumerate(new_position_results):

            bee_to_compare = current_positions[idx]
            if bee_to_compare.is_better_food(result[1]):
                if bee._is_employed:
                    next_bee_generation.append(Bee(
                        result[0],
                        result[1],
                        self._stay_limit,
                        True
                    ))
                else:
                    next_bee_generation.append(Bee(
                        result[0],
                        result[1],
                        self._stay_limit,
                        False
                    ))
            else:
                next_bee_generation.append(bee_to_compare)

        self._bees = next_bee_generation
            
    def fit(self, n_iter, threshold, verbose=True):
        print("Initializing Bee Colony...")
        self.init_colony()
        from tqdm.notebook import tqdm
        print("Optimizing...")
        for i in tqdm(range(n_iter)):
            best_bee = self.best()
            if i % 2 == 0 and verbose:
                print(f'Best Score: {best_bee._score}')
                print(f'Best Params: {best_bee._params}')
            if best_bee._score > threshold:
                break
            self.next_generation()
        print(f'Best Score: {best_bee._score}')
        print(f'Best Params: {best_bee._params}')

    # def 

def rmse_wf_validation(model, params, series, transformer):
    wf_valid, wf_pred = walkforward_validation(model, params, series, gap=2, step=4, verbose=False)
    return -mean_squared_error(reverse_transform_series(wf_valid, transformer), reverse_transform_series(wf_pred, transformer), squared=False)

# We can save the trained model clf using pickle
import pickle

# warnings.filterwarnings("ignore")
tuned_ABCs = []
for i, model_setting in enumerate(all_model_settings[0:1]):
#     pipeline = make_pipeline(ct, RandomOverSampler(random_state=42), model_setting["estimator"]())
    # (self, estimator, search_space, common_params, fitness_fn, fitness_fn_params, n_employed_bees, n_processes, stay_limit=20, mutation_chance=0.5, mutation_std=0.05, scoring_metric="ap")
    tuned_ABCs.append(ArtificalBeeColony(
        model_setting["tsa_model"],
        electricity_consumption_series,
        model_setting["tunable_space"],
        model_setting["common_parameters"],
        model_setting["default_hyperparameters"],
        rmse_wf_validation,
        {},
        electricity_pt,
        10,
        1,
        stay_limit=1,
        mutation_chance=0.6,
        mutation_std=0.4,
        baseline_score=-1000,
        ignore_warnings=False
        ))
    tuned_ABCs[i].fit(100, 1000)
    pickle.dump(tuned_ABCs[i].best(), open(f'pickled/ABC/Gas Consumption/{model_setting["tsa_model"](gas_consumption_series, {}).__class__.__name__}.pkl', 'wb'))

Initializing Bee Colony...
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-158.91590139482443
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'heuristic'}
-113.51658753536212
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 46, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'estimated'}
-150.03233056986903
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': None, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-149.91607453825222
Optimizing...

  0%|          | 0/100 [00:00<?, ?it/s]

Best Score: -113.51658753536212
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'heuristic', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-153.96410601775906
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 6, 'initialization_method': 'estimated'}
-47.69325240072313
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 36, 'initialization_method': 'estimated'}
-50.38136469558918
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 42, 'initialization_method': 'estimated'}
-64.26466300599563
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'heuristic'}
-147.9059621500737
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 46, 'initialization_method': 'estimated'}
-157.2241064914079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-155.8615167447571
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 33, 'initialization_method': 'estimated'}
-146.16436989956333
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-158.91590139482443
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 14, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 2, 'initialization_method': 'heuristic'}
-149.76571432743057
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-171.15954994922163
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 16, 'initialization_method': 'heuristic'}
-151.43827428544267
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-149.24749601182177
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'heuristic'}
-149.4013762306907
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'heuristic'}
-130.2042276987179
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-149.24749601182177
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'heuristic'}
-113.51658753536212
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 11, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'heuristic'}
-113.51658753536212
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-142.81666654837161
Best Score: -42.56571522808076
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 42, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'heuristic'}
-148.04152863153863
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 19, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-155.01196579268253
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-157.10340129014205
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 3, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'estimated'}
-143.65264109388195
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 7, 'initialization_method': 'estimated'}
-150.10265372219234
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 31, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-151.7080483247815
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-149.24674036314093
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'heuristic'}
-152.64978188332157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-153.96410601775906
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-42.56571522808076
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-160.26832736853328
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'estimated'}
-159.70360231569668
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-149.4013762306907
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-148.5680959328079
Best Score: -42.56571522808076
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-150.37229450203748
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 9, 'initialization_method': 'estimated'}
-147.95640635242648
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-154.97441174619894
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-153.2004549933681
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 28, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 40, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 42, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-142.81666654837161
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-170.04909121560405
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-42.56571522808076
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 6, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'estimated'}
-152.64978188332157
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-152.81774952450633
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 5, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 3, 'initialization_method': 'estimated'}
-146.44004090776565
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'heuristic'}
-45.46185364463555
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-155.8615167447571
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-42.56571522808076
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 48, 'initialization_method': 'estimated'}
-45.46185364463555
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 40, 'initialization_method': 'heuristic'}
-160.13967890272886
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'estimated'}
-149.24404357875906
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 22, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 21, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-160.26832736853328
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 22, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 27, 'initialization_method': 'heuristic'}
-154.2288764010476
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 21, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-155.09298548824927
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 7, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-149.66802942502787
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'heuristic'}
-148.04152863153863
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 3, 'initialization_method': 'estimated'}
-146.44004090776565
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'heuristic'}
-147.9059621500737
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 46, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 3, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 8, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'heuristic'}
-155.09298548824927
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-152.81774952450633
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-128.2550032692836
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-57.20584677533819
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-113.51658753536212
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 22, 'initialization_method': 'estimated'}
-149.21253320008157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-142.81666654837161
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 22, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'heuristic'}
-152.64978188332157
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 8, 'initialization_method': 'heuristic'}
-150.4485959421343
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 7, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-156.84909053419952
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-149.24749601182177
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-155.01196579268253
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 17, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-160.26832736853328
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-64.11276326883838
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-153.96410601775906
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 3, 'initialization_method': 'heuristic'}
-145.17426150217804
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 12, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 29, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-148.80266939468228
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 47, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 42, 'initialization_method': 'heuristic'}
-64.26466300599563
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 44, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 9, 'initialization_method': 'heuristic'}
-147.95640635242648
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'heuristic'}
-154.0311939068065
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 15, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-158.37608596662065
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 8, 'initialization_method': 'heuristic'}
-149.24454578740762
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 2, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 6, 'initialization_method': 'heuristic'}
-47.69325240072313
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-160.26832736853328
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'estimated'}
-152.21121301304237
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-149.55198541246693
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-142.81666654837161
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 37, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-154.2288764010476
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-147.13517349487515
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 40, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 3, 'initialization_method': 'estimated'}
-145.17426150217804
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-147.9059621500737
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 38, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 18, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 19, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'estimated'}
-152.64978188332157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-148.35780762233975
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-159.3614020506685
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-148.5680959328079
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-170.04909121560405
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 14, 'initialization_method': 'heuristic'}
-149.24749601182177
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 4, 'initialization_method': 'estimated'}
-148.80266939468228
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-148.80266939468228
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 36, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-113.51658753536212
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-155.23027240287004
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'estimated'}
-149.24404357875906
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-156.84909053419952
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-153.59694645963435
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-148.04152863153863
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-150.7770577465336
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-151.43827428544267
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 6, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 44, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 5, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-150.7770577465336
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-149.4013762306907
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'estimated'}
-153.2004549933681
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 25, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 5, 'initialization_method': 'heuristic'}
-149.17028691400697
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-160.426588485646
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-158.91590139482443
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 5, 'initialization_method': 'estimated'}
-150.50756110046356
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'heuristic'}
-149.55198541246693
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 2, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 25, 'initialization_method': 'heuristic'}
-128.2550032692836
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-154.97441174619894
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-153.96410601775906
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-138.96711970144207
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-149.4013762306907
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'heuristic'}
-147.9059621500737
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'heuristic'}
-159.70360231569668
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 47, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 47, 'initialization_method': 'estimated'}
-104.61049629534872
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'estimated'}
-149.24674036314093
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-170.04909121560405
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-41.592876695913674
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 22, 'initialization_method': 'estimated'}
-149.21253320008157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 22, 'initialization_method': 'heuristic'}
-149.21253320008157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-57.20584677533819
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-58.342072274053194
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 5, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-155.8615167447571
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 9, 'initialization_method': 'heuristic'}
-147.95640635242648
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 6, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 5, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'heuristic'}
-150.37229450203748
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 27, 'initialization_method': 'heuristic'}
-154.2288764010476
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 10, 'initialization_method': 'estimated'}
-149.4013762306907
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 30, 'initialization_method': 'heuristic'}
-64.88471474874893
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 39, 'initialization_method': 'heuristic'}
-154.64820031234552
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-160.26832736853328
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'estimated'}
-159.3614020506685
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-170.04909121560405
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 7, 'initialization_method': 'estimated'}
-150.10265372219234
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 36, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'estimated'}
-149.24674036314093
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'heuristic'}
-149.24404357875906
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 38, 'initialization_method': 'heuristic'}
-154.47153385934928
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 7, 'initialization_method': 'heuristic'}
-150.10265372219234
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 32, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'heuristic'}
-155.01196579268253
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'heuristic'}
-150.03233056986903
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 2, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-113.51658753536212
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-149.24749601182177
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'estimated'}
-152.64978188332157
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-153.2004549933681
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 43, 'initialization_method': 'heuristic'}
-152.6019006999114
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-152.21121301304237
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 19, 'initialization_method': 'heuristic'}
-149.24404357875906
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-151.7080483247815
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 47, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-130.2042276987179
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 45, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 12, 'initialization_method': 'estimated'}
-41.33967086938133
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-146.16436989956333
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-151.6991011083506
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 14, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 39, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 20, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 19, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-171.15954994922163
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 25, 'initialization_method': 'heuristic'}
-138.96711970144207
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-148.04152863153863
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 32, 'initialization_method': 'heuristic'}
-160.26832736853328
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-150.7770577465336
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'heuristic'}
-149.91607453825222
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 37, 'initialization_method': 'estimated'}
-152.64978188332157
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 13, 'initialization_method': 'heuristic'}
-143.65264109388195
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-153.2004549933681
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 6, 'initialization_method': 'estimated'}
-47.69325240072313
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 38, 'initialization_method': 'estimated'}
-154.47153385934928
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 29, 'initialization_method': 'heuristic'}
-157.10340129014205
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-151.7080483247815
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-150.37229450203748
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-64.11276326883838
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'estimated'}
-147.13517349487515
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 20, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 23, 'initialization_method': 'estimated'}
-151.7080483247815
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-157.10340129014205
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'estimated'}
-158.1150359021839
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 15, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'heuristic'}
-171.15954994922163
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 12, 'initialization_method': 'heuristic'}
-41.33967086938133
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'estimated'}
-156.84909053419952
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 2, 'initialization_method': 'estimated'}
-148.0786351728811
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 12, 'initialization_method': 'estimated'}
-41.90794225620957
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 6, 'initialization_method': 'heuristic'}
-46.670623189128115
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 29, 'initialization_method': 'estimated'}
-157.10340129014205
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-64.88471474874893
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 22, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 5, 'initialization_method': 'estimated'}
-149.17028691400697
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 16, 'initialization_method': 'heuristic'}
-151.43827428544267
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 11, 'initialization_method': 'estimated'}
-130.2042276987179
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 42, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-150.37229450203748
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-155.01196579268253
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-149.24674036314093
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 23, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 17, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'heuristic'}
-158.1150359021839
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 27, 'initialization_method': 'estimated'}
-154.2288764010476
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-155.8615167447571
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'heuristic'}
-151.6991011083506
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 44, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 41, 'initialization_method': 'heuristic'}
-153.2004549933681
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 38, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 31, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'heuristic'}
-150.7770577465336
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 4, 'initialization_method': 'heuristic'}
-149.66802942502787
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 16, 'initialization_method': 'estimated'}
-150.31129738474286
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 36, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 35, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-154.0311939068065
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 34, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'heuristic'}
-151.6991011083506
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 12, 'initialization_method': 'estimated'}
-148.5680959328079
Best Score: -40.731873326949376
Best Params: {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 39, 'initialization_method': 'estimated'}
-154.64820031234552
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 18, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 46, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 6, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'heuristic'}
-147.13517349487515
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 33, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'estimated'}
-158.1150359021839
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 2, 'initialization_method': 'heuristic'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 28, 'initialization_method': 'heuristic'}
-148.04152863153863
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 26, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 21, 'initialization_method': 'heuristic'}
-148.35780762233975
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-153.59694645963435
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-154.0311939068065
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-41.592876695913674
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-151.6991011083506
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 22, 'initialization_method': 'heuristic'}
-149.21253320008157
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 22, 'initialization_method': 'heuristic'}
-150.00831090683047
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 2, 'initialization_method': 'estimated'}
-149.76571432743057
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 45, 'initialization_method': 'heuristic'}
-158.91590139482443
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 31, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 18, 'initialization_method': 'estimated'}
-57.20584677533819
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 5, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 31, 'initialization_method': 'heuristic'}
-153.59694645963435
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 17, 'initialization_method': 'estimated'}
-155.09298548824927
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 30, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 8, 'initialization_method': 'heuristic'}
-149.91607453825222
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 20, 'initialization_method': 'estimated'}
-151.6991011083506
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'heuristic'}
-40.731873326949376
{'trend': 'add', 'damped_trend': False, 'seasonal': None, 'seasonal_periods': 15, 'initialization_method': 'estimated'}
-149.91607453825222
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': True, 'seasonal': None, 'seasonal_periods': 24, 'initialization_method': 'estimated'}
-148.5680959328079
{'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 25, 'initialization_method': 'estimated'}
-138.96711970144207
{'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 35, 'initialization_method': 'estimated'}

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Cell In[235], line 26
      6 for i, model_setting in enumerate(all_model_settings[0:1]):
      7 #     pipeline = make_pipeline(ct, RandomOverSampler(random_state=42), model_setting["estimator"]())
      8     # (self, estimator, search_space, common_params, fitness_fn, fitness_fn_params, n_employed_bees, n_processes, stay_limit=20, mutation_chance=0.5, mutation_std=0.05, scoring_metric="ap")
      9     tuned_ABCs.append(ArtificalBeeColony(
     10         model_setting["tsa_model"],
     11         electricity_consumption_series,
   (...)
     24         ignore_warnings=False
     25         ))
---> 26     tuned_ABCs[i].fit(100, 1000)
     27     pickle.dump(tuned_ABCs[i].best(), open(f'pickled/ABC/Gas Consumption/{model_setting["tsa_model"](gas_consumption_series, {}).__class__.__name__}.pkl', 'wb'))

Cell In[227], line 386, in ArtificalBeeColony.fit(self, n_iter, threshold, verbose)
    384     if best_bee._score > threshold:
    385         break
--> 386     self.next_generation()
    387 print(f'Best Score: {best_bee._score}')
    388 print(f'Best Params: {best_bee._params}')

Cell In[227], line 299, in ArtificalBeeColony.next_generation(self)
    294             new_onlooker_results.append(bee_pool.apply_async(
    295                 self.score,
    296                 [neighbor_food]
    297             ))
    298         else:
--> 299             new_onlooker_results.append(self.score(
    300                 neighbor_food
    301             ))
    303 else:
    305     current_positions.append(bee)

Cell In[227], line 84, in ArtificalBeeColony.score(self, params)
     82                 return (params, self._baseline_score)
     83         else:
---> 84             score = self._fitness_fn(self._tsa_model, params, self._series, self._transformer)
     85 #             print("bruh")
     86             print(score)

Cell In[228], line 2, in rmse_wf_validation(model, params, series, transformer)
      1 def rmse_wf_validation(model, params, series, transformer):
----> 2     wf_valid, wf_pred = walkforward_validation(model, params, series, gap=2, step=4, verbose=False)
      3     return -mean_squared_error(reverse_transform_series(wf_valid, transformer), reverse_transform_series(wf_pred, transformer), squared=False)

Cell In[106], line 77, in walkforward_validation(model, params, data, rolling, gap, step, forecast_period, validation_size, flatten, return_train, return_coefficients, verbose)
     74         tsa_model = model(train_sets[i], params)
     75 #         print(train_sets[i].index)
     76 #         print(len(train_sets[i]), len(train_sets[i]))
---> 77         tsa_model_fit = tsa_model.fit()
     78         forecast = tsa_model_fit.predict(start=len(train_sets[i]), end=len(train_sets[i])+forecast_period-1)
     79         forecast_sets.append(forecast)

File ~\anaconda3\lib\site-packages\pandas\util\_decorators.py:211, in deprecate_kwarg.<locals>._deprecate_kwarg.<locals>.wrapper(*args, **kwargs)
    209     else:
    210         kwargs[new_arg_name] = new_arg_value
--> 211 return func(*args, **kwargs)

File ~\anaconda3\lib\site-packages\pandas\util\_decorators.py:211, in deprecate_kwarg.<locals>._deprecate_kwarg.<locals>.wrapper(*args, **kwargs)
    209     else:
    210         kwargs[new_arg_name] = new_arg_value
--> 211 return func(*args, **kwargs)

File ~\anaconda3\lib\site-packages\pandas\util\_decorators.py:211, in deprecate_kwarg.<locals>._deprecate_kwarg.<locals>.wrapper(*args, **kwargs)
    209     else:
    210         kwargs[new_arg_name] = new_arg_value
--> 211 return func(*args, **kwargs)

File ~\anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:1141, in ExponentialSmoothing.fit(self, smoothing_level, smoothing_trend, smoothing_seasonal, damping_trend, optimized, remove_bias, start_params, method, minimize_kwargs, use_brute, use_boxcox, use_basinhopping, initial_level, initial_trend)
   1139 method = "SLSQP" if method is None else method
   1140 if optimized:
-> 1141     res = self._optimize_parameters(
   1142         res, use_brute, method, minimize_kwargs
   1143     )
   1144 else:
   1145     l0, b0, s0 = self.initial_values(
   1146         initial_level=initial_level, initial_trend=initial_trend
   1147     )

File ~\anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:837, in ExponentialSmoothing._optimize_parameters(self, data, use_brute, method, kwargs)
    833 bounds = np.array(orig_bounds[:3], dtype=float)
    834 hw_args = HoltWintersArgs(
    835     sel.astype(int), params, bounds, y, m, self.nobs
    836 )
--> 837 params = self._get_starting_values(
    838     params,
    839     start_params,
    840     use_brute,
    841     sel,
    842     hw_args,
    843     bounds,
    844     init_alpha,
    845     func,
    846 )
    848 # We always use [0, 1] for a, b and g and handle transform inside
    849 mod_bounds = [(0, 1)] * 3 + orig_bounds[3:]

File ~\anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:743, in ExponentialSmoothing._get_starting_values(self, params, start_params, use_brute, sel, hw_args, bounds, alpha, func)
    741 best_params = points[0]
    742 for point in points:
--> 743     val = opt(point, hw_args)
    744     if val < best_val:
    745         best_params = point

File ~\anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:82, in opt_wrapper.<locals>.f(*args, **kwargs)
     80 def f(*args, **kwargs):
     81     err = func(*args, **kwargs)
---> 82     if isinstance(err, np.ndarray):
     83         return err.T @ err
     84     return err

KeyboardInterrupt: 

Hyperparameter Tuning Results: SARIMA

print("Gas Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Gas Consumption/SARIMAX.pkl', 'rb'))._params)

print("\nElectricity Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Electricity Consumption/SARIMAX.pkl', 'rb'))._params)

print("\nWater Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Water Consumption/SARIMAX.pkl', 'rb'))._params)

Gas Consumption Parameters: 
 {'trend': 'c', 'measurement_error': True, 'simple_differencing': False, 'enforce_stationarity': False, 'enforce_invertibility': True, 'trend_offset': 4, 'order': (0, 1, 2), 'seasonal_order': (0, 0, 1, 14)}

Electricity Consumption Parameters: 
 {'trend': 'c', 'measurement_error': True, 'simple_differencing': False, 'enforce_stationarity': False, 'enforce_invertibility': False, 'trend_offset': 2, 'order': (1, 1, 1), 'seasonal_order': (0, 1, 4, 12)}

Water Consumption Parameters: 
 {'trend': 'n', 'measurement_error': True, 'simple_differencing': False, 'enforce_stationarity': True, 'enforce_invertibility': True, 'trend_offset': 4, 'order': (1, 1, 2), 'seasonal_order': (2, 0, 0, 12)}

tsa_scores["gas"] = tsa_scores["gas"].append(evaluation_descriptive(gas_consumption_series, sarimax, pickle.load(open(f'pickled/ABC/Gas Consumption/SARIMAX.pkl', 'rb'))._params, gas_pt, return_scores=True))
tsa_scores["electricity"] = tsa_scores["electricity"].append(evaluation_descriptive(electricity_consumption_series, sarimax, pickle.load(open(f'pickled/ABC/Electricity Consumption/SARIMAX.pkl', 'rb'))._params, electricity_pt, return_scores=True))
tsa_scores["water"] = tsa_scores["water"].append(evaluation_descriptive(water_consumption_series, sarimax, pickle.load(open(f'pickled/ABC/Water Consumption/SARIMAX.pkl', 'rb'))._params, water_pt, return_scores=True))

plot_model_params(
    gas_consumption_series, 
    sarimax, 
    pickle.load(open(f'pickled/ABC/Gas Consumption/SARIMAX.pkl', 'rb'))._params, 
    gas_pt, 
    title='Gas Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

plot_model_params(
    electricity_consumption_series, 
    sarimax, 
    pickle.load(open(f'pickled/ABC/Electricity Consumption/SARIMAX.pkl', 'rb'))._params, 
    electricity_pt, 
    title='Electricity Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

plot_model_params(
    water_consumption_series, 
    sarimax, 
    pickle.load(open(f'pickled/ABC/Water Consumption/SARIMAX.pkl', 'rb'))._params, 
    water_pt, 
    title='Water Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=96,
    test_length=24
)

MAE (test): 2.003144560332195
RMSE (test): 2.359555900434219

MAE (test): 28.617484520697413
RMSE (test): 40.56621676179449

MAE (test): 85.82681093800795
RMSE (test): 106.59544264041853

Hyperparameter Tuning Results: Holt-Winter's Exponential Smoothing

print("Gas Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Gas Consumption/HWES.pkl', 'rb'))._params)

print("\nElectricity Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Electricity Consumption/HWES.pkl', 'rb'))._params)

print("\nWater Consumption Parameters: \n", pickle.load(open(f'pickled/ABC/Water Consumption/HWES.pkl', 'rb'))._params)

Gas Consumption Parameters: 
 {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 36, 'initialization_method': 'estimated', 'use_boxcox': False}

Electricity Consumption Parameters: 
 {'trend': 'add', 'damped_trend': True, 'seasonal': 'add', 'seasonal_periods': 24, 'initialization_method': 'estimated', 'use_boxcox': False}

Water Consumption Parameters: 
 {'trend': 'add', 'damped_trend': False, 'seasonal': 'add', 'seasonal_periods': 44, 'initialization_method': 'estimated', 'use_boxcox': False}

tsa_scores["gas"] = tsa_scores["gas"].append(evaluation_descriptive(gas_consumption_series, hwes, {}, gas_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))
tsa_scores["electricity"] = tsa_scores["electricity"].append(evaluation_descriptive(electricity_consumption_series, hwes, {}, electricity_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))
tsa_scores["water"] = tsa_scores["water"].append(evaluation_descriptive(water_consumption_series, hwes, {}, water_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))

AIC:  -239.64375381156736

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  2.4080061700599367
MAE (train):  0
RMSE (test):  3.1185787723068548
RMSE (train):  0
AIC:  -535.3806058283955

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  116.17916650440775
MAE (train):  0
RMSE (test):  138.68556400848195
RMSE (train):  0
AIC:  -193.58420943996398

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  106.4568994846988
MAE (train):  0
RMSE (test):  130.2133239901612
RMSE (train):  0

print("MAE (test):  2.4080061700599367\nRMSE (test):  3.1185787723068548")

print("MAE (test):  116.17916650440775\nRMSE (test):  138.68556400848195")

print("MAE (test):  106.4568994846988\nRMSE (test):  130.2133239901612")

MAE (test):  2.4080061700599367
RMSE (test):  3.1185787723068548
MAE (test):  116.17916650440775
RMSE (test):  138.68556400848195
MAE (test):  106.4568994846988
RMSE (test):  130.2133239901612

plot_model_params(
    gas_consumption_series, 
    hwes, 
    {}, 
    gas_pt, 
    title='Gas Consumption Untuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

plot_model_params(
    electricity_consumption_series, 
    hwes, 
    {},
    electricity_pt, 
    title='Electricity Consumption Untuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

plot_model_params(
    gas_consumption_series, 
    hwes, 
    {}, 
    water_pt, 
    title='Water Consumption Untuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

MAE (test): 2.283070889330169
RMSE (test): 2.799318605517012

MAE (test): 152.38749810969824
RMSE (test): 176.3203566348815

MAE (test): 59.05157788424297
RMSE (test): 72.7657631048426

tsa_scores["gas"] = tsa_scores["gas"].append(evaluation_descriptive(gas_consumption_series, hwes, pickle.load(open(f'pickled/ABC/Gas Consumption/HWES.pkl', 'rb'))._params, gas_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))
tsa_scores["electricity"] = tsa_scores["electricity"].append(evaluation_descriptive(electricity_consumption_series, hwes, pickle.load(open(f'pickled/ABC/Electricity Consumption/HWES.pkl', 'rb'))._params, electricity_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))
tsa_scores["water"] = tsa_scores["water"].append(evaluation_descriptive(water_consumption_series, hwes, pickle.load(open(f'pickled/ABC/Water Consumption/HWES.pkl', 'rb'))._params, water_pt, return_scores=True, plot_arima=False, return_coefficients=False, is_hwes=True))

AIC:  -193.06607679378027

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  2.061169582476098
MAE (train):  0
RMSE (test):  2.685972340559238
RMSE (train):  0
AIC:  -1316.043576320356

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  30.17105853506808
MAE (train):  0
RMSE (test):  39.66503384845227
RMSE (train):  0
AIC:  -152.8129216617915

  0%|          | 0/15 [00:00<?, ?it/s]

MAE (test):  97.27641004265811
MAE (train):  0
RMSE (test):  120.11915522640722
RMSE (train):  0

plot_model_params(
    gas_consumption_series, 
    hwes, 
    pickle.load(open(f'pickled/ABC/Gas Consumption/HWES.pkl', 'rb'))._params, 
    gas_pt, 
    title='Gas Consumption Tuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

plot_model_params(
    electricity_consumption_series, 
    hwes, 
    pickle.load(open(f'pickled/ABC/Electricity Consumption/HWES.pkl', 'rb'))._params,
    electricity_pt, 
    title='Electricity Consumption Tuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

plot_model_params(
    gas_consumption_series, 
    hwes, 
    pickle.load(open(f'pickled/ABC/Water Consumption/HWES.pkl', 'rb'))._params, 
    water_pt, 
    title='Water Consumption Tuned HWES Forecast',
    past_length=96,
    test_length=24,
    is_hwes=True,
    plot_conf_int=False
)

MAE (test): 2.1364532730625334
RMSE (test): 2.607434383313296

MAE (test): 33.85792224857919
RMSE (test): 46.01787454310246

MAE (test): 65.3553433660503
RMSE (test): 78.35565997882517

tsa_scores["gas"].index = ["base_sarima", "auto_arima", "base_varmax", "tuned_base_sarima", "tuned_hwes", "hwes"]
tsa_scores["electricity"].index = ["base", "tuned_base_sarima", "tuned_hwes", "hwes"]
tsa_scores["water"].index = ["base_sarima", "auto_arima", "base_varmax", "tuned_auto_arima", "tuned_hwes", "hwes"]

print("Gas Consumption:")

tsa_scores["gas"]

Gas Consumption:

	mae_test	mae_train	rmse_test	rmse_train
base_sarima	2.348216	2.485460	3.027845	3.704156
auto_arima	2.371986	2.398876	3.032369	3.574951
base_varmax	2.649908	14.978199	3.287332	18.214568
tuned_base_sarima	2.357685	2.469937	2.996810	3.634326
tuned_hwes	2.061170	0.000000	2.685972	0.000000
hwes	2.408006	0.000000	3.118579	0.000000

print("Electricity Consumption:")

tsa_scores["electricity"]

Electricity Consumption:

	mae_test	mae_train	rmse_test	rmse_train
base	27.227981	19.465739	34.916816	28.718052
tuned_base_sarima	26.118219	20.574414	34.027548	30.522336
tuned_hwes	30.171059	0.000000	39.665034	0.000000
hwes	116.179167	0.000000	138.685564	0.000000

print("Water Consumption:")

tsa_scores["water"]

Water Consumption:

	mae_test	mae_train	rmse_test	rmse_train
base_sarima	86.987716	74.458845	107.624587	102.813271
auto_arima	82.213413	73.323133	102.176881	101.657298
base_varmax	60.964439	782.540704	76.329591	1006.868281
tuned_auto_arima	80.065222	73.617893	99.721392	101.602199
tuned_hwes	97.276410	0.000000	120.119155	0.000000
hwes	106.456899	0.000000	130.213324	0.000000

Final Models

HWES is the best performing for Gas Consumption and SARIMA is the best performing for Electricity Consumption.

Even though VARMAX, for Water Consumption, performed better than the rest of the models in terms of the test RMSE and MAE, it fails by comparison in terms of its training score. Plotting the in-sample and out-of-sample forecasts also resulted in a very irregular plot. Thus, I choose SARIMA in place of it.

plot_model_params(
    gas_consumption_series, 
    hwes, 
    pickle.load(open(f'pickled/ABC/Gas Consumption/HWES.pkl', 'rb'))._params, 
    gas_pt, 
    title='Gas Consumption Tuned HWES Forecast',
    past_length=300,
    test_length=1,
    forecast_length=6,
    is_hwes=True,
    plot_conf_int=False,
    figsize=(18, 6)
)

plot_model_params(
    electricity_consumption_series, 
    sarimax, 
    pickle.load(open(f'pickled/ABC/Electricity Consumption/SARIMAX.pkl', 'rb'))._params, 
    electricity_pt, 
    title='Electricity Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=300,
    forecast_length=6,
    test_length=1,
    figsize=(18, 6)
)

plot_model_params(
    water_consumption_series, 
    sarimax, 
    pickle.load(open(f'pickled/ABC/Water Consumption/SARIMAX.pkl', 'rb'))._params, 
    water_pt, 
    title='Water Consumption SARIMA Forecast with 95% Confidence Interval',
    past_length=300,
    test_length=1,
    forecast_length=6,
    figsize=(18, 6)
)

MAE (test): 0.8161350210288632
RMSE (test): 0.8161350210288632

MAE (test): 94.56872272531518
RMSE (test): 94.56872272531518

MAE (test): 145.1563526096636
RMSE (test): 145.1563526096636